# On finite capable $p$-group of class two

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

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 $2$.
• $\textrm{SmallGroup}(64,91)$ is of class 3. – A.G Mar 13 '16 at 11:03
• There is no example $H$ of order $512$ or less. – verret Mar 13 '16 at 23:16