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I'm looking for a group $G$ that is not nilpotent but has a nilpotent Frattini quotient $G/\Phi(G)$. Such a group would have to be necessarily infinite.

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Thompson's group F. – user641 Dec 3 '12 at 9:26
I think it's also true in the Grigorchuk group. The maximal subgroups all have index 2, and $G/\Phi(G)$ is elementary abelian of order 8. – Derek Holt Dec 3 '12 at 9:47

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