TITLE: Bijection between irreducible representations and conjugacy classes of finite groups
QUESTION [33 upvotes]: Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?

REPLY [4 votes]: It appears  that similar question has been asked at sci.math.research  Tue, 19 Oct 1999. The answer by G. Kuperberg is quite interesting. Hope no one don't mind if I put it here:

As Torsten Ekedahl explained, it is sometimes the wrong question, but
  in modified form, the answer is sometimes yes.
For example, consider A_5, or its central extension Gamma = SL(2,5).
  The two 3-dimensional representations are Galois conjugates and there
  is no way to choose one or the other in association with the conjugacy
  classes.  However, if you choose an embedding pi of Gamma in SU(2),
  then there is a specific bijection given by the McKay correspondence.
  The irreducible representations form an extended E_8 graph where two
  representations are connected by an edge if you can get from one to
  the other by tensoring with pi.  The conjugacy classes also form and
  E_8 graph if you resolve the singularity of the algebraic surface
  C^2/Gamma.  The resolution consists of 8 projective lines intersecting
  in an E_8 graph.  If you take the unit 3-sphere S^3 in C^2, then the
  resolution gives you a surgery presentation of the 3-manifold
  S^3/Gamma. The surgery presentation then gives you a presentation of
  Gamma itself called the Wirtinger presentation.  As it happens, each
  of the Wirtinger generators lies in a different non-trivial conjugacy
  class. In this way both conjugacy classes and irreps. are in bijection
  with the vertices of E_8.