Why do sporadic simple groups have so few conjugacy classes? In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have.  So it is to be expected that non-abelian finite simple groups have a smallish number of conjugacy classes relative to size.  Even with this intuition though, some of the numbers appearing in the list of sporadic simple groups are a bit surprising.  For instance, the Monster group, with more than $10^{53}$ elements and $15$ prime divisors, has fewer than $200$ conjugacy classes.  (An alternating group of comparable order has more than $30 000$ conjugacy classes.)  $M_{22}$ has $443520$ elements and only $12$ conjugacy classes.
What is going on here?  Is there something about the special combinatorial structures that allow these groups to exist that also makes the centralisers exceptionally small?
 A: This isn't really an answer, but an extended comment that might shed a little light.
Let $k(G)$ be the number of conjugacy classes in a group $G$, and let me interpret your question as a query about the proportion $k(G)/|G|$. The following paper suggests that the behaviour you describe for sporadic groups is actually shared by the finite groups of Lie type too.... but that the proportion $k(G)/|G|$ "takes longer" to reach zero.

Liebeck, Martin W.; Pyber, László Upper bounds for the number of conjugacy classes of a finite group. J. Algebra 198 (1997), no. 2, 538–562. 

The main result of this paper is that $k(G)\leq (6q)^{\ell}$ for a group of Lie type of untwisted rank $\ell$ over a field of $q$ elements. 
In the case, for instance, where $G={\rm PSL}_{\ell+1}(q)$, we know that $|G|>q^{\frac12\ell^2}$ for $q$ big enough, so one obtains immediately that (for $\ell>2$) $\lim\limits_{|G|\to\infty} \frac{k(G)}{|G|}=0$. 
When $\ell=1, 2$, one can check directly that the same is true.
Indeed, for the other families, one has similar lower bounds on the size of $|G|$, except that one might need to replace the $\frac12$ by some other constant. Thus, in general, it is true that for $G$ in the family of simple groups of Lie type 
$$\lim\limits_{|G|\to\infty} \frac{k(G)}{|G|}=0.$$
Thus the groups of Lie type are exhibiting exactly the same behaviour asymptotically as what you observe for sporadic groups.
Of course, I have chosen just one interpretation of your question, and even from this perspective, this is only a partial answer: I cannot explain why the sporadic groups reach such small values for $\frac{k(G)}{|G|}$ so much more quickly than the groups of Lie type. In truth, though, I doubt an "answer" such as this really exists, since saying anything about the general behaviour of sporadic groups seems incredibly difficult.
Added later: Actually, I just looked at the ATLAS and I must say I don't see a great deal of evidence that the sporadic simple groups in general have much fewer conjugacy classes than groups of Lie type of similar sizes. For instance:


*

*$k(M_{24})=26, \, \, k(G_2(4))=32$

*$k(HN)=54, \, \, k(Fi_{22})=65, \, \, k(F_4(2))=95$

*$k(O'N)=39,  \, \, k(C0_3)=42, \, \, k(O_{10}^+(2))=97$


In each row, groups have comparable size. In each case the sporadic group(s) have a few less conjugacy classes but not by a great deal. Perhaps your question should really be about the Monster?
