The maximum k-plex problem (MKPP) is an significant relaxation version of the maximum clique problem with extensive applications. Recently, lots of researchers have proposed many heuristic algorithms based on various methods to solve the MKPP. In this work, to further improve the performance of solving the MKPP, we propose an efficient local search algorithm based on three main ideas. First, we propose a relaxed bounded configuration checking strategy that considers two kinds of historical searching information to relax the restricted strength of configuration checking and the forbidden condition of candidate vertices for the operation Add, respectively. Second, we present a novel solution information-based vertex selection strategy based on two kinds of solution information to select high-quality candidate vertices. Third, we define the solution core and then introduce a core-based perturbation strategy to help the algorithm jump out of local optima. The experimental results show that the proposed algorithm significantly outperforms the state-of-the-art MKPP algorithms in almost all the instances.