Conjugate type vectors and commuting probability in a finite group

In this paper, we first introduce some classes of fine non-abelian metacyclic 2-groups of nilpotency class two and nilpotency class at least three, which have been classified by Beuerle. Then, we define all representations of these groups as a finite non abelian two generators 2-group, namely Ɠ, so that this group covers all possible cases of the Beuerle’s classifications. Next, we compute the centralizers and the order of the centralizers of elements in the group Ɠ. Moreover, the conjugate type vector of the group, is obtained. Finally, as a direct application of the results, explicit formulas for the th commutativity degree of such groups are given. As a conclusion, it is found at the sizes of all the finite non-abelian metacyclic 2- groups of each type in Ɠ are the same. Also, we observe that the th commutativity degree of Ɠ has the same formula for all the group presentations.