A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$.

Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\Phi(G)$ is elementary abelian $p$-group of order $p^n$. As an example of such a group we can give $G=\underbrace{\mathbb{Z}\_{p^{2}}\times\mathbb{Z}\_{p^{2}}\times\dots\times\mathbb{Z}\_{p^{2}}}\_{n\,\,times}$. Does there exist any nonabelian $p$-group with mentioned properties?