Let $G$ be a finite group. Define $\Phi_{-}(G)$ as the subgroup of $G$ generated by all the minimal subgroups of $G$ (a minimal subgroup of $G$ is a subgroup of $G$ of prime order).
It is easy to check that the subgroup $\Phi_{-}(G)$ is normal in $G$. I think that the quotient group $G/\Phi_{-}(G)$ is nilpotent but I don't see how to prove it.