Do there exists a finite capable $p$-group $G$ of class two with cyclic center and the center is not subgroup of Frattini subgroup of $G$?
A group $G$ is capable if there exists a group $H$ such that $G\cong\dfrac{H}{Z(H)}$.
2026-03-26 19:01:07.1774551667
On finite capable $p$-group of class two
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The smallest examples have order $256$ and there are a few of those. For example, for $H=\textrm{SmallGroup}(256,4509)$, we have $H/Z(H)=G\cong \textrm{SmallGroup}(64,91)$ and $G$ has cyclic center of order $4$ which intersects the Frattini subgroup of $G$ in a subgroup of order $2$.