Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?



p-groups and 2-generated abelian images
Let $G$ be any finite $p$-group, whose center and derived subgroup both have order $p$. Then every proper quotient of $G$ is abelian. In particular, if $G$ is not generated by 2 elements, it answers the question.


For instance, let $H$ be any non-abelian group of order $p^3$, denote by $Z$ its center. Define $G$ as the quotient of $H\times H$ by the diagonal of $Z\times Z$. Then $G$ has order $p^5$ satisfies the above conditions; its minimal number of generators is 4.


