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.