Large vertex-symmetric Higman graphs with $\mu=6$

A strongly regular graph with $v={m\choose 2}$ and $k=2(m-2)$ is called a Higman graph. In such a graph, the parameter $\mu$ is 4, 6, 7, or 8. If $\mu=6$, then $m\in\{9,17,27,57\}$. Vertex-symmetric Higman graphs were classified by N.D. Zyulyarkina and A.A. Makhnev (all of these graphs turned out to have rank 3). A program of classification of vertex-symmetric Higman graphs is implemented. Earlier Zyulyarkina and Makhnev found vertex-symmetric Higman graphs with $\mu=6$ and $m\in\{9,17\}$. In the present paper, vertex-symmetric Higman graphs with $\mu=6$ and $m\in{27,57}$ are studied. It is interesting that the group $G/S(G)$ may contain two components $L$ and $M$. In the case $m=27$, we have $M\cong A_5,A_6$ and $L\cong L_3(3)$; in the case $m=57$, we have either $M\cong PSp_4(3)$ and $L\cong L_3(7)$ or $M\cong A_6$ and $L\cong J_1$.