Finite $p$-groups having Schur multiplier of maximum order

Abstract: Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ denote the Schur multiplier of $G$. Niroomand proved that $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)+1}$ for non-abelian $p$-groups $G$ of order $p^n$ with derived subgroup of order $p^k$. Recently Rai classified $p$-groups $G$ of nilpotency class $2$ for which $|M(G)|$ attains this bound. In this article we show that there is no finite $p$-group $G$ of nilpotency class $c \geq 3$ for $p\neq3$ such that $|M(G)|$ attains this bound. Hence $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)}$ for $p$-groups $G$ of class $c \geq 3$ where $p \neq 3$. We also construct a $p$-group $G$ for $p=3$ such that $|M(G)|$ attains the Niroomand's bound.