Are two groups isomorphic iff their cycle index is the same? Note that for every group there exists a permutation group to which it is isomorphic.
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No, as per Graphical Enumeration by Hararay and Palmer, p.37:
This is a quote from a translation of Polya, I can give the full citation if you want (it's in German).
Note that the term "identical," in this context is isomporphism and isomorphism is homomorphism (the reference text is very old).