  We determine that there is no partial geometry ${\cal G}$ with parameters
$(s,t,\alpha)=(4,27,2)$. The existence of such a geometry has been a
challenging open problem of interest to researchers for almost 40 years. The
particular interest in ${\cal G}$ 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.
