Aiming at systematically correcting the non-Galilean-invariant thermal diffusivity in the previous multiple-relaxation-time Boltzmann equation collision model [Shan and Chen, Int. J. Mod. Phys. C 18, 635 (2007)IJMPEO0129-183110.1142/S0129183107010887], we show that by separately relaxing the central moments of the distribution function, Chapman-Enskog calculation leads to the correct hydrodynamic equations with mutually independent and Galilean invariant viscosity and thermal diffusivity, provided the velocity-space discretization preserves moments up to the fourth order. By transforming the central moments back to the absolute reference frame and evaluating using fixed discrete velocities, the efficient and accurate streaming-collision time-stepping algorithm is preserved. The lattice Boltzmann model is found to have excellent numerical stability in high-Reynolds-number simulations.