Automorphisms of a distance-regular graph with intersection array {35, 32, 28; 1, 4, 8}

We continue the study of automorphisms of distance-regular locally cyclic graphs with at most 4096 vertices (the intersection arrays of such graphs were found earlier by A.A. Makhnev and M.S. Nirova). Let $\Gamma$ be a distance-regular graph with intersection array $\{35,32,28;1,4,8\}$. Then it has eigenvalue $\theta_2=-1$ and the graph $\bar \Gamma_3$ is pseudogeometric for the net $pG_8(35,8)$ and has parameters $(1296,315,90,72)$. We study possible automorphisms of such graphs. In particular, for a graph $\Gamma$ with intersection array $\{35,32,28;1,4,8\}$ and $G={\rm Aut}(\Gamma)$, it is proved that $\pi(G)\subseteq \{2,3,5,7\}$. Further, if a nonsolvable group $G={\rm Aut}(\Gamma)$ acts transitively on the vertex set of a graph with intersection array $\{35,32,28;1,4,8\}$ and $\bar T$ is the socle of the group $\bar G=G/S(G)$, then $G=S(G)G_a$, $\bar T_a\cong A_5$, and $\bar T_{a,b}\cong A_4$ for some vertices $a\in \Gamma$ and $b\in [a]$.