Let $G$ be a real semisimple Lie group. Really, I only care about $\text{SL}(n,\mathbb{R})$ and $\text{Sp}(2n,\mathbb{R})$.

I'd like to perform a computer search for a finite group with a certain property (that property is not important here). To do this, I need to be able to efficiently compute the isomorphism classes (as Lie groups) of the centralizers of certain finite subgroups of $G$. The finite groups involved might be rather large.

This brings me to two related questions. First, where can I find algorithms for this? Second, what would be the best form in which to generate my finite subgroups? For instance, is it fine to just give generators, should I give a list of all the matrices in the finite group, do I also need a presentation of the finite groups, etc?