David Unaipon (1872-1967) is celebrated on the $50 note. As he was from South Australia he may have been able to vote in the first Federal election under the law of that state. He died in the year of the race power referendum which he championed. Not until 1962 were Indigenous people entitled to vote in Federal elections. 

Voting rights for indigenous people in Australia.

the 1962 amendment to the electoral act

It is important to understand the history of the vote for indigenous people in Australia from colonial times to the present day.

In March 1962 the Commonwealth Electoral Act was amended to provide that Indigenous people could enrol to vote in federal elections if they wished. Unlike other Australians it was not compulsory for them to enrol. It was also an offence for anyone to use undue influence or pressure to induce them to enrol. Once they enrolled, however, voting was compulsory.

Having the right to vote federally was one thing, it took longer for reforms to be made regarding the mechanics of voter registration for indigenous people in the states.

The 1967 referendum

Getting the vote preceded the famous 1967 Constitutional Referendum.

The 1967 Referendum approved two changes to the Constitution affecting Aboriginal people.

The first change was to one of the heads of power in section 51 of the Constitution to allow the Commonwealth to take responsibility for indigenous issues. This amended clause xxvi to read:

The people of any race for whom it is deemed necessary to make special laws.

The words removed were -  'other than the aboriginal race in any State'

Although sometimes trumpeted as a triumph of rights, this part of the 1967 referendum gave the Commonwealth a malignant authority to use the race power unchecked - as discussed by George Williams in 2012 referring to the High Court case Kartinyeri v Commonwealth (1998) 195 CLR 337 (The Hindmarsh Island Bridge Case).

On the afternoon of the first day of the hearing, the Commonwealth Solicitor-General, Gavan Griffith, suggested that the races power "is infused with a power of adverse operation". He acknowledged "the direct racist content of this provision‟ in the sense of "a capacity for adverse operation‟. The following exchange then occurred:
Kirby J: Can I just get clear in my mind, is the Commonwealth's submission that it is entirely and exclusively for the Parliament to determine the matter upon which special laws are deemed necessary … or is there a point at which there is a justiciable question for the Court? I mean, it seems unthinkable that a law such as the Nazi race laws could be enacted under the race power and that this Court could do nothing about it.
Griffith QC: Your Honour, if there was a reason why the Court could do something about it, a Nazi law, it would, in our submission, be for a reason external to the races power. It would be for some wider over-arching reason.
Williams adds, '"Of course, without a Bill of Rights or express protection from racial discrimination, there was no such over-arching reason."

The second change was to remove Section 127 which was headed "Aborigines not to be counted in reckoning population":

In reckoning the numbers of the people of the Commonwealth, or of a State or other part of the Commonwealth, aboriginal natives shall not be counted.

Being counted in the population had a special meaning specific to voting because the numbers of people of the Commonwealth in each state affected the quota for the allocation of House of Representatives seats.  

should there be dedicated seats for indigenous people?

Dr John Chesterman wrote in 2006 about the tangible legal possibility of creating designated seats for Indigenous representation in the Federal Parliament. And the Indigenous Law Centre, based at UNSW Law, has a page of links to resources relevant to the possible creation of Dedicated Indigenous Parliamentary Seats

Constitutional recognition

The Recognise movement is campaigning for Constitutional recognition and removal of racist provisions in the Constitution being sections 51 (26) (as discussed above) and section 25 which says:

25. Provisions as to races disqualified from voting
For the purposes of the last section, if by the law of any State all persons of any race are disqualified from voting at elections for the more numerous House of the Parliament of the State, then, in reckoning the number of the people of the State or of the Commonwealth, persons of that race resident in that State shall not be counted.

Recognise puts the case for change:

That referendum in 1967 was a watershed moment for Australia, deleting two racially discriminatory references. Yet it did not complete the constitutional task of securing equality. Two further sections of the Constitution – sections 25 and 51(xxvi) – still permit racial discrimination by governments. As constitutional scholars have noted, Australia is now the only democratic nation in the world with a Constitution that still authorises race discrimination.
The nation’s founding document also still includes no recognition of Aboriginal and Torres Strait Islander people and the fact that our country’s history spans tens of thousands of years prior to the British colonies being proclaimed.
It implicitly begins Australia’s national story only from British arrival.
As Harold Ludwick, a Bulgun Warra man from Cape York, puts it: “If the Constitution was the birth certificate of Australia, we’re missing half the family”.
We need to fix this, and bring the country together after so many chapters apart. It is the next step in reconciling our past. And it’s the right thing to do.
Australians also know that our country’s unique Aboriginal and Torres Strait Islander cultures enrich this nation, and are crucial to our distinctive national identity. The success of constitutional recognition can help to protect against the loss of our unique Indigenous cultures for future generations.

Gender and ethnicity issues

The Australian Human Rights Commission argues for constitutional reform concerning indigenous recognition.

Mala Htun considered in 2004 the intersection of remedies proposed to overcome underrepresentation of women and of ethnic groups in electoral democracies: 

Some 50 countries officially allocate access to political power by gender, ethnicity, or both. Yet in the world's electoral democracies, the policies used for women differ systematically from those used for ethnic groups. The former receive candidate quotas in parties; the latter, reserved seats in legislatures. Why? My explanation focuses on the varying ways that gender and ethnic identities intersect with partisan cleavages and on the distinct "work" performed by the different remedies for underrepresentation. Quotas, which make space within existing parties, are appropriate for groups whose boundaries crosscut partisan divisions. Reservations, which create incentives for the formation of group-specific parties and permit them direct representation, suit groups whose boundaries coincide with political cleavages. Since gender is crosscutting while ethnicity tends to be coinciding, women receive candidate quotas while ethnic groups get legislative reservations. Claims for inclusion via quotas pose less of a challenge to liberal institutions than claims to difference through legislative reservations. Case studies of representational politics in France, India, and Peru illustrate the argument.

Issues of gender and race come together when considering female underrepresentation in indigenous representative bodies. This was addressed in 2009 by Megan Davis writing 'A woman's place...' in the Griffith Review. Talking about the then Aboriginal and Torres Strait Islander Commission (ATSIC) Davis points to the virtue of mandating equal gender representation in the indigenous representative body:

Improving women's representation by mandated equal representation should not be viewed as a discriminatory act towards men, but rather a temporary form of substantive equality to help women achieve political representation. Whatever the outcome of discussions about a national representative body, decision-makers must be cognisant of the fact that any institution will have a different impact upon Indigenous men and women. ATSIC prioritised the advancement of the ‘rights agenda' as a major platform which masked the gendered effects on women's already disadvantaged position. One example is the conflict between the right to sustain a customary practice and the right of women and children to live free from violence.