  In 2018 the first, Rukavina and the third author constructed with the aid of
a computer the first example of a strongly regular graph $\Gamma$ with
parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant
vertex-transitive graph on 216 vertices. In this paper, using the geometry of
the Hermitian surface of PG(3, 4), we provide a computer-free proof of the
existence of the graph $\Gamma$. The maximal cliques of $\Gamma$ are also
determined.
