Class $\mathcal{S}$ superconformal indices from maximal supergravity

We present a new gauging of maximal supergravity in five spacetime dimensions with gauge group containing ISO(5), involving the local scaling symmetry of the metric, and admitting a supersymmetric anti-de Sitter vacuum. We show this maximal supergravity to arise by consistent truncation of M-theory on the (non-spherical, non-parallelisable) six-dimensional geometry associated to a stack of $N$ M5-branes wrapped on a smooth Riemann surface. The existence of this truncation allows us to holographically determine the complete, universal spectrum of light operators of the dual four-dimensional $\mathcal{N}=2$ theory of class $\mathcal{S}$. We then compute holographically the superconformal index of the dual field theory at large-$N$, finding perfect agreement with previously known field theory results in specific limits.