I've recently plunged into understanding the basics of group theory, mostly out of sheer fascination, and all sorts of interesting questions are coming to mind. Please remember I'm new at this; if your response to "Do not bite the tyro" is "A true mathematician does not coddle the weak", please stop reading here and move on. You'll save us both a lot of irritation.
I am curious what rules exist to determine whether two groups could be isomorphic, based on their Cayley tables. I'm familiar already with counting the number of elements of order two (How to determine if groups are isomorphic using cayley table) but wonder if any other "tricks" are known that can reduce the amount of brute force searching for possible isomorphisms.
More precisely: If I have two Cayley tables S and T, and want to know if there is a permutation P that will take S to T, we know the answer is No if S and T have different numbers of elements of order 2. Are there any other tricks that tell us there is no such permutation; or, better yet, tell us that there certainly is such a permutation, without having to do a brute search for the permutation?