We describe two methods for computing the low-dimensional integral homology
of the Mathieu simple groups and use them to make computations such as
$H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via
Sylow subgroups. The other method uses a Wythoff polytope and perturbation
techniques to produce an explicit free $\ZZ M_n$-resolution. Both methods apply
in principle to arbitrary finite groups.