In this talk I will talk about the classification problem for automorphism groups of cubic fourfolds. Cubic fourfold is important because of its close relation with hyper-Kahler geometry and rationality problems. Via Hodge theory, the study of automorphism groups of cubic fourfolds can be transferred into the study of groups of Hodge isometries of some weight two Hodge structures. A key observation is that those groups can be detected from Leech lattice, the unique unimodular even definite rank 24 lattice with no roots. The main result (joint with Radu Laza) is a full classification of symplectic automorphism groups of smooth cubic fourfolds. There are 34 such groups, all being subgroups of the Conway group.