  We compute the small cohomology ring of the Cayley Grassmannian, that
parametrizes four-dimensional subalgebras of the complexified octonions. We
show that all the Gromov-Witten invariants in the multiplication table of the
Schubert classes are non negative and deduce Golyshev's conjecture O holds true
for this variety. We also check that the quantum cohomology is semisimple and
that there exists, as predicted by Dubrovin's conjecture, an exceptional
collection of maximal length in the derived category.
