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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 $G\cong\dfrac{H}{Z(H)}$.

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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 $2$.

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  • $\begingroup$ $\textrm{SmallGroup}(64,91)$ is of class 3. $\endgroup$ – A.G Mar 13 '16 at 11:03
  • $\begingroup$ Sorry, I missed that requirement. $\endgroup$ – verret Mar 13 '16 at 11:05
  • $\begingroup$ There is no example $H$ of order $512$ or less. $\endgroup$ – verret Mar 13 '16 at 23:16

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