Strongly regular graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$

In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $\Gamma$ can be used to define a new strongly regular graph $\widetilde{\Gamma}$, where the vertices of the graph $\widetilde{\Gamma}$ correspond to the orbits of $\Gamma$ (the rows of $M$). We show that some of the obtained graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.