Is the following group either a quaternion group or $D_8$?

Question

Let $|G|=2^n$ and $Z(G)=G'=\Phi(G)$ where $\Phi(G)$ is the Frattini subgroup and $|Z(G)|=2$.

Is $G$ necassarily either a quaternion group or $D_8$?

Answer 1

Those conditions imply that $G$ is a central product of copies of $D_8$ and $Q_8$ with amalgamated centre, and groups of this type are called extraspecial. So we have $n=2k+1$, where $k$ is the number of copies.

It can be shown that the central product $Q_8 \circ Q_8$ of $Q_8$ and $Q_8$ is isomorphic to $D_8 \circ D_8$, but not to $Q_8 \circ D_8$. So, for each $k \ge 1$, there are two isomorphism types of extraspecial groups of order $2^{2k+1}$.

A similar situation applies with $p$ replaced by an odd prime $p$: there are two isomorphism types of extraspecial groups of order $p^{2k+1}$ but, unlike in the case $p=2$, one of these types has exponent $p$.