This paper investigates group consensus control for first-order multi-agent systems (MASs) with input saturation and intermittent communication. A novel class of distributed group consensus protocols is designed by combining current and outdated states of agents, which differs from the standard one using only current states. Sufficient conditions for group consensus are derived for MASs without two conservative assumptions required by previous literature. The proposed protocol can accelerate the convergence speed of MASs by choosing appropriate outdated states. Several examples are provided to illustrate the effectiveness of our results.