With Ike Feitler‘s help:
According to Wolfram Alpha, is
arc-transitive, biconnected, bridgeless, cage, Cayley graph, chromatically unique, circulant, claw-free , complete, connected, cyclic, determined by spectrum, distance-regular, distance-transitive, edge-transitive, Hamilton-connected, Hamiltonian, Hamming, integral, Johnson, Kneser, LCF, line graph, nonplanar, perfect, regular, strongly regular, symmetric, traceable, Turán, unitransitive, and vertex-transitive. (links will come with a future edit to this post).
It has graph spectrum and 666 (spooky!) edges. The graph stability index is 1.203 trillion.