Let \(G\) be a finite group with center \(Z(G)\), and let \(x_1,\ldots,x_k\) be representatives of the conjugacy classes of \(G\) that contain more than one element. For \(x\in G\), let \(C_G(x) \coloneqq \theset{ g\in G \st gxg^{-1} = x }\) be the centralizer of \(x\). Then \[ \abs{G} = \abs{Z(G)} + \sum_{i=1}^k [G: C_G(x_i)]. \]
\(G\) acts on itself by conjugation, \(g\cdot x=gxg^{-1}\); its orbits are the conjugacy classes, which partition \(G\), and the stabilizer of \(x\) is \(C_G(x)\). By the orbit-stabilizer theorem, the conjugacy class of \(x\) has \([G:C_G(x)]\) elements. The class of \(x\) is \(\theset{x}\) exactly when \(gxg^{-1}=x\) for all \(g\in G\), that is, when \(x\in Z(G)\). Summing class sizes, the singleton classes contribute \(\abs{Z(G)}\) and the remaining classes contribute \(\sum_{i=1}^k[G:C_G(x_i)]\).