TITLE: Why do sporadic simple groups have so few conjugacy classes?
QUESTION [35 upvotes]: 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?

REPLY [18 votes]: This is also rather an expanded comment. --
Since for purely arithmetical reasons, $\ln(\ln(|G|))$ is a lower bound
for the number $k(G)$ of conjugacy classes of a finite group $G$, maybe
$$
  f(G) := \ln(k(G))/\ln(\ln(\ln(|G|)))
$$
is a better measure than $k(G)/|G|$ for how many or how few conjugacy
classes a group $G$ has in comparison with its order. For example we have
(examples ordered by group order):

$f({\rm A}_5) \approx 4.68799$,
$f({\rm PSL}(2,7)) \approx 3.64930$,
$f({\rm A}_6) \approx 3.39930$,
$f({\rm PSL}(2,8)) \approx 3.64187$,
$f({\rm PSL}(2,11)) \approx 3.3204$,
$f({\rm A}_7) \approx 3.04392$,
$f({\rm PSL}(3,3)) \approx 3.23520$,
$f({\rm M}_{11}) \approx 2.92936$,
$f({\rm PSL}(2,31)) \approx 3.53994$,
$f({\rm A}_8) \approx 3.17897$,
$f({\rm PSL}(3,4)) \approx 2.77366$,
$f({\rm Sz}(8)) \approx 2.83466$,
$f({\rm M}_{12}) \approx 3.03727$,
$f({\rm J}_1) \approx 2.96687$,
$f({\rm A}_9) \approx 3.16283$,
$f({\rm M}_{22}) \approx 2.63787$,
$f({\rm J}_2) \approx 3.20085$,
$f({\rm A}_{10}) \approx 3.23851$,
$f({\rm M}_{23}) \approx 2.76986$,
$f({\rm A}_{11}) \approx 3.31013$,
$f({\rm Sz}(32)) \approx 3.39405$,
$f({\rm HS}) \approx 3.01600$,
$f({\rm J}_3) \approx 2.88256$,
$f({\rm M}_{24}) \approx 3.00146$,
$f({\rm PSL}(6,2)) \approx 3.55208$,
$f({\rm O'N}) \approx 2.85566$,
$f({\rm Fi}_{22}) \approx 3.36345$,
$f({\rm HN}) \approx 3.18141$,
$f({\rm B}) \approx 3.54764$,
$f({\rm A}_{43}) \approx 6.61233$,
$f({\rm M}) \approx 3.34883$,
$f({\rm A}_{44}) \approx 6.69491$.