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. 



Cohomologically trivial $G$-modules
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.


