A tensor product approach to compute 2-nilpotent multiplier of p-groups

Let G be a group given by a free presentation G similar or equal to F/R. The 2-nilpotent multiplier of G is the abelian group M-(2)( G) = (R boolean AND gamma(3)(F))/ gamma(3)(R, F) which is invariant of G [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc. 58 (1945) 295-419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405-428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of M-(2)(G), when G is a finite (generalized) extra special p-group. Moreover, the descriptions of the triple tensor product circle times(3)G, and the triple exterior product Lambda(3)G are given.