  We classify the symplectic automorphism groups for cubic fourfolds. The main
inputs are the global Torelli theorem for cubic fourfolds and the
classification of the fixed-point sublattices of the Leech lattice. Among the
highlights of our results, we note that there are 34 possible groups of
symplectic automorphisms, with 6 maximal cases. The six maximal cases
correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of
them previously identified by other authors. Finally, the Fermat cubic fourfold
has the largest possible order (174,960) for the automorphism group
(non-necessarily symplectic) among all smooth cubic fourfolds.
