Zbl 1227.20014. of A. Jaikin-Zapirain paper says that there is conjecture by Ya. Berkovich and V. Zhmud that: **number of conj classes > number of prime factos of G** (achieved for M_22 and PSL(3,4) )

Quote from Andrei Jaikin-Zapirain paper (Adv. Math. 227, No. 3, 1129-1143 (2011). )

**Conjecture**. There exists a constant C> 0
such that any finite group G of order n satisfies k(G) > C log2( n).

Main theorem of this paper is the following:

In this paper we establish the first
super-logarithmic lower bound for the
number of conjugacy classes of a finite
nilpotent group.

Theorem 1.1. There exists a
(explicitly computable) constant C > 0
such that every finite nilpotent group
G of order n 8 satisfies

k(G) > C log2 (n) ((log2 log2 n) / ( log2 log2 log2 n) )

Introduction to the paper contains discussion of some history of the subject is quite readable.

On p-groups having the minimal number of conjugacy classes of maximal size
A. Jaikin-Zapirain, M. F. Newman and E. A. O’Brien

A long-standing question is the
following: do there exist p-groups of
odd order having precisely p-1
conjugacy classes of the largest
possible size? We exhibit a 3-group
with this property.

http://springerlink.com/content/n737pq17655582qv
1970 The number of conjugacy classes in a finite group. Patrick X. Gallagher.

This paper contains results comparing number of conjugacy classes in a group and in its subgroup.

http://arxiv.org/abs/1102.4107

Multiplicities of conjugacy class sizes of finite groups

Hung Ngoc Nguyen

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Zaikin-Zapirain. In this note, we prove that if $G$ is a finite simple group then the order of $G$, denoted by $|G|$, is bounded in terms of the largest multiplicity of its conjugacy class sizes and that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group $G$ is $m$, then $|G|$ is bounded in terms of $m$.