We consider a linear-quadratic-Gaussian (LQG) game with a major player and a large number of minor players with mean field coupling. The state of the major player appears in the dynamics of the minor players, causing the mean field to be responsive to its control. We construct decentralized ε-Nash strategies. This is accomplished by combining (i) stochastic control with random coefficients and (ii) a procedure called anticipative variational calculations which addresses the major player's ability to simultaneously perturb its own state process and the mean field.