Finite p-groups having Schur multiplier of maximum order

Let G be a non-abelian p-group of order pn and M(G) denote the Schur multiplier of G. Niroomand proved that |M(G)|≤p12(n+k−2)(n−k−1)+1 for non-abelian p-groups G of order pn with derived subgroup of order pk. 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≥3 for p≠3 such that |M(G)| attains this bound. Hence |M(G)|≤p12(n+k−2)(n−k−1) for p-groups G of class c≥3 where p≠3. We also construct a p-group G for p=3 such that |M(G)| attains the Niroomand's bound.