Irreducible characters separate conjugacy classes ($(\forall \chi \in \mathrm{Irr}(G),\chi (g)=\chi (h))⟺g{\sim }_{G}h$)

The irreducible characters of a finite group form a basis of its complex class functions. Equality of every irreducible character value therefore forces equality of every class-function value, including indicators of conjugacy classes, and hence conjugacy. The converse follows from the class-function property.