On group characterization by numbers of conjugate classes

Let $c(n,G)$ be a number of conjugate elements of order n in a group $G.$ In the article we study the problem of recognition of finite group by the set $\mathrm{ncl}(G)$ that consists of numbers $c(n,G).$ We prove that Abelian groups can be recognized by the set $\mathrm{ncl}(G)$ when the order of the group is known. We also describe some other types of groups that can be recognized. The examples of non-isomorphic groups with the same sets $\mathrm{ncl}(G)$ are given. Some theorems about a group recognition by partial conditions on $c(n,G)$ are proved.