Among the given options, choose the odd one out.
$$27,125,343,512,729$$

Let's express each number as a perfect cube:
* $27=3^3$ (Cube of an odd number)
* $125=5^3$ (Cube of an odd number)
* $343=7^3$ (Cube of an odd number)
* $512=8^3$ (Cube of an even number)
* $729=9^3$ (Cube of an odd number)

Since $512$ is the only cube of an even number in the list, it is the odd one out.