A note on the Schur multiplier of groups of prime power order

By a well-known result of Green (Proc R Soc A 237:574-581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191-196, 1999), there is an integer t, say corank(G), such that vertical bar M(G)vertical bar = p(1/2 n(n-1)-t). In Niroomand (J Algebra 322:4479-4482, 2009), the author showed for a non-abelian group G, corank(G) >= log(p) (vertical bar G vertical bar) - 2 and classified the structure of all non-abelian p-groups of corank log(p) (vertical bar G vertical bar) - 2. In the present paper, we are interesting to characterize the structure of all p-groups of corank log(p) (vertical bar G vertical bar) - 1.