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In this paper, we prove that the maximum k-club problem (MkCP) defined on an undirected graph is NP-hard. We also give an integer programming formulation for this problem as well as an exact branch-and-bound algorithm and computational results on instances involving up to 200 vertices. Instances defined on very dense graphs can be solved to optimality within insignificant computing times. When k=2, the most difficult cases appear to be those where the graph density is around 0.15.