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Let $G$ be a finite nilpotent nonabelian group. Is it true that for every natural number $k$ there exists a finite group $G_k$ such $G_k$ is not isomorphic to a subgroup of a direct power of $G$ while every $k$-generated subgroup of $G_k$ is isomorphic to such a subgroup.

I know that for abelian groups this is not possible.

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This is true and has been proved in Olʹšanskiĭ, A. Ju. Conditional identities in finite groups. Sibirsk. Mat. Ž. 15 (1974), 1409–1413. In fact he proved a similar fact for any finite group with a non-abelian Sylow subgroup. Moreover if all Sylow subgroups are abelian, the fact is no longer true.

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