An algorithm for finding a maximum clique in an arbitrary graph is described. The algorithm has a worst-case time bound of $k(1.286)^{n}$ for some constant $k$, where $n$ is the number of vertices in the graph. Within a fixed time, the algorithm can analyze a graph with 2 3/4 as many vertices as the largest graph which the obvious algorithm (examining all subsets of vertices) can analyze. Keywords: Algorithm, Clique, Graph, Independent Set.