# Complexity of finite group isomorphism problem

Consider the next decision problem:
Given two finite groups represented by their multiplicity table, determine if they are isomorphic or not.
Clearly, this problem belongs to NP since given a witness in the form of an isomorphism between the groups it can be verified in polynomial time (to the input size). But what more can be said about this problem in terms of complexity?

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–  Colin McQuillan Aug 12 '12 at 20:57
Interesting. Thank you. –  Ido Aug 12 '12 at 21:06
Good question. I'd be interested to know more about the hardest case: p-groups. –  user641 Aug 12 '12 at 22:53
Also, are things easier when the groups are permutation groups? –  user641 Aug 12 '12 at 22:54