Given a simple undirected graph G, the maximum k-club problem is to find a maximum-cardinality subset of nodes inducing a subgraph of diameter at most k in G. This NP-hard generalization of clique, originally introduced to model low diameter clusters in social networks, is of interest in network-based data mining and clustering applications. We give two MAX-SAT formulations of the problem and show that two exact methods resulting from our encodings outperform significantly the state-of-the-art exact methods when evaluated both on sparse and dense random graphs as well as on diverse real-life graphs from the literature.