We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.