There exists a unique (smooth) cubic hypersurface of dimension 7 in $\mathbb{P}^8$ which is invariant under the order 3420 simple group $\mathrm{PSL}(2,19)$.
We study the intermediate Jacobian of that hypersurface and we prove that its subjacent 85-dimensional torus has a structure of an Abelian variety.
We also study a natural 9-dimensional abelian variety with $\mathrm{PSL}(2,19)$ action, which is related to the intermediate Jacobian.