If the factor group with respect to the center of $G$ is cyclic, then $(aZ(G))^n=gZ(G)$ for some $n$ and any $g$, where both $a$ and $g$ are from $G$ (and $a^n$ is, too).
Because of the definition of the operation on the factor group it should be correct that $a^nZ(G)=gZ(G)$.
How come $G$ is not cyclic, too?