Prove that a finite group of 8

**A**
The number of conjugacy classes of a finite group *G*
equals the number of irreducible representations of *G*.
The order of *G* is the sum of the squares of the dimensions
of these representations. At least one of these dimensions is 1
(the trivial representation). When |*G*|=8*k*,
it is not possible to write |*G*|-1=8*k*-1
as a sum of fewer than four squares.
Therefore the number of representations,
which is also the number of conjugacy classes,
is at least 4+1=5, QED.

*Remarks*: Equality holds for at least three groups *G*,
namely the dihedral and quaternion groups of order 8, and the fourth
symmetric group, of order 4!=24. Michael Larsen observed
that for each *m* there is an effective upper bound
on the size of a finite group with at most *m* conjugacy classes;
when *m*=5, this reduces the problem, and the determination
of the cases of equality, to a finite (though possibly impractical)
computation. The upper bound is a consequence of the “class equation”:
|*G*| is the sum of the sizes of its conjugacy classes,
each of which is a factor of |*G*|, and at least one of which
equals 1.