While this might sound like a simple question, the answer is a mess of possible options, plus sub-variants. It depends on what what "centre" means, and "of where".
Unlike a regular shape such as a circle or a square, Australia (or almost any land area) is an untidy tangle of bays, inlets, peninsulas and islands.
Should we offend Tasmanians or ignore Macquarie Island and the Antarctic territories? Does the coastline follow high or low tide?
And, if you really want to get confused, there's an island off the coast of Dubai called "Australia"! (Along with a plethora of others such as "Somalia", "Malaysia" and "Pakistan".)
This problem is acknowledged by Australia's chief mapping authority, Geoscience Australia: "Officially, there is no centre of Australia. This is because there are many complex but equally valid methods... "
They list five centres that are based only on the Australian mainland.
These include the "Centre of Gravity", "Lambert Gravitational Centre", "Furthest point from the coastline", "Geodetic Median Point" and the "Johnston Geodetic Station".
They're spread across nearly 400 kilometres of central/south Northern Territory, which gives them only vague agreement on our question.
That suggests a meta-method that aggregates these points into what we might call the "Centre of Centres", but again, we have to decide how to calculate that.
Whatever, a loose answer places it at 135 kilometres south-west of Alice Springs.
A very different - literally human centred - approach is based on population. By that method, the centre is about 35 kilometres east of the NSW town of Ivanhoe.
What all this demonstrates is the often necessarily arbitrary, value-based decisions that underlie statistics. While a line on a map clinically dissects two regions, the reality is very different for people who live near borders. Their culture and social connections rarely fit neatly on one side or the other.
For Australians there's another centre that is highly significant and is not previously acknowledged.
Pub Dead Centre is the location on the continent most distant from the nearest pub. This is the place from which you are furthest from a cold beer.
Pub Dead Centre is about 50 kilometres east of the Gary Highway (one of Len Beadell's "bomb" roads) in the Gibson Desert.
While it is harsh and beautiful country, it is a sad, sad place if you didn't bring an esky.
However, even that is unimportant compared to another.
In this pivotal moment in the nation's history, the heart of Australia is in Uluru.
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