This paper introduces and studies the maximum $k$-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a $k$-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.