The nonabelian tensor square G⊗G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg′ ⊗ h = (gg′ ⊗ h)(g ⊗ h) and g ⊗ hh′ = (g ⊗ h)(g ⊗ hh′) for all g, g, h, h′ ∈ G, where gg′ = gg′g−1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the m...

Using a new classification of 2-generator p-groups of class 2, we compute various homological functors for these groups. These functors include the nonabelian tensor square, nonabelian exterior square and the Schur multiplier. We also determine which of these groups are capable and which are unicentral.

‎Let $G$ be a finite $p$-group of order $p^n$ and‎ ‎$|{mathcal M}(G)|=p^{frac{1}{2}n(n-1)-t(G)}$‎, ‎where ${mathcal M}(G)$‎ ‎is the Schur multiplier of $G$ and $t(G)$ is a nonnegative integer‎. ‎The classification of such groups $G$ is already known for $t(G)leq‎ ‎6$‎. ‎This paper extends the classification to $t(G)=7$.