My knowlege in group theory is very limited and this question is out of pure curiousity:
Given a random finite group, how many conjugacy classes will it probably have?
I can try to make this question more precise:


Define $a\_n$ to be the average number of conjugacy classes in finite groups of order less or equal $n$.


What is the asymptotic behaviour of the sequence $a\_n$?



Number of conjugacy classes in generic finite group?
In elementary terms, you have to analyze the following class equation
$ n = 1 + h\_2 + ... + h\_r $
where


* n is the order of the group G
* $h\_k$ denotes the number of elements in the k-th conjugacy class, and $ n = c\_k.h\_k$.


Dividing by n, you get
$1 = \frac{1}{n} + \frac{1}{c\_2} + ... + \frac{1}{c\_r} $ which has a finite number of solutions.


Christine Ayoub in her paper *On the number of conjugate classes in a group* (Proc. Internat. Conf Theory of Groups Canberra 1967) has worked out this analysis for p-groups and there are probably more recent papers on this aspect, which Scott and others allude to in the comments. See for example


1. *MR2557143 Keller, Thomas Michael .
Lower bounds for the number of
conjugacy classes of finite groups.
Math. Proc. Cambridge Philos. Soc. 
147 (2009), no. 3, 567--577.*


Another way of looking at your question is to see that the number of conjugacy classes is the same as the number of irreducible representations. The [character table](http://en.wikipedia.org/wiki/Character_table) is always square. Therefore, one could ask "*what are the number of irreducible characters Irr(G) in a finite group of order n?*". The number of linear characters are [G:G'] where G'=[commutator subgroup](http://en.wikipedia.org/wiki/Commutator_subgroup) but the nonlinear ones are tougher and there are papers establishing various bounds for these.


1. *MR2526321 (2010d:20010) Aziziheris,
Kamal ; Lewis, Mark L. Counting
the number of nonlinear irreducible
characters of a finite group. Comm.
Algebra 37 (2009), no. 5,
1572--1578.*
2. *MR0689258 (84d:20014) Wada,
Tomoyuki . On the number of
irreducible characters in a finite
group. Hokkaido Math. J. 12 
(1983), no. 1, 74--82.*
3. *MR0798751 (87a:20006) Wada,
Tomoyuki . On the number of
irreducible characters in a finite
group. II. Hokkaido Math. J. 14 
(1985), no. 2, 149--154.*


