The Variety of Flexes of Plane Cubics

Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with a faithful algebraic action of $\mathrm {PSL}_3$, and (2) $X$ is $\mathrm {PSL}_3$-equivariantly birationally isomorphic to a homogeneous fiber space over $\mathrm {PSL}_3/K$ with fiber $\mathbb P^1$ for some subgroup $K$ isomorphic to the binary tetrahedral group $\mathrm {SL}_2(\mathbb F_3)$.