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Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle \prod_{i=1}^{2^d} a^{x_i} \;|\; a\in Z(\Phi(G)) \rangle,$$ where $\{x_1,\dots,x_{2^d} \}$ is a right transversal of $\Phi(G)$ in $G$?

This means that $Z(\Phi(G))$ is a cohomologically trivial $G/\Phi(G)$-module.

Here $\Phi(G)$ is the Frattini subgroup of $G$ and $Z(H)$ denotes the center of a group $H$.

We know that such groups of order $2^8$ exist and exactly 10 groups of order $2^8$ with the above property exist.

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The answer is no. I did a computer search through the $10494213$ groups of order $512$ (which you could have done yourself) and found that the only group that satisfies the condition is the elementary abelian group.

I did successfully confirm that there are $10$ examples of order $2^8$, in addition to the elementary abelian group.

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  • $\begingroup$ Many thanks. I could not carry all groups of order 512, but my colleague Maria Guedri was running a program under GAP to do it. She has ruled out about 1/3 of such groups or more. $\endgroup$ – Alireza Abdollahi Apr 14 '16 at 12:13

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