Wythoff polytopes and low-dimensional homology of Mathieu groups

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as H5(M23,Z)=Z7 and H3(M24,Z)=Z12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free ZMn-resolution. Both methods apply in principle to arbitrary finite groups.