The number of conjugacy classes in a finite nilpotent group (Q1072644)

Let G be a finite group of order \(p^ m\) (p prime) with centre Z(G) of order \(p^ b\) and let r(G) be the number of conjugacy classes of G. This paper contains a number of interesting equations, inequalities and congruences relating r(G) to other invariants of G. A principal and representative result is the following one. Suppose G has a maximal abelian subgroup A of order \(p^ a\). Then there exists an integer \(k\geq 0\) such that \[ r(G)=(p^{2a}/p^ m)+(p^ b(p+1)(p^{m-a}-1)/p^{m- a})+k(p^ 2-1)(p-1)/p^{m-a}. \]