A characterization of finite $p$-groups by their Schur multiplier

Abstract: Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of order $p^n$ for $t(G)=\log_p(|G|)+1$.