A result of Hurwitz is that the number of automorphisms of a Riemann surface or complex algebraic curve of genus $g geq 2$ is bounded by $84(g-1)$. In genus 7, there is a curve called the Fricke-Macbeath curve that achieves this bound. We discuss sets of equations of this curve due to Macbeath and Brock as well as a new set of equations that arise by representing this curve in Mukai's model of $overline{M}_7$.