Software or tool for investigating groups

Question

I'm interested in software that has the ability to investigate finite groups. In particular, I'd like to be able to ask it questions like "What are the solutions to $x^3 = 1$?" (i.e. find cube roots of unity) for a permutation group.

I've had a look at Maxima but don't see a way to ask it do that.

Answer 1

The canonical anwer is GAP.

(You can use it from inside Sage)

Answer 2

There is also Magma, but unlike GAP it is not (in general) free.