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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_3$; (2) $X$ is ${\mathrm{P}\mathrm{S}\mathrm{L}}_3$-equivariantly birationally isomorphic to a homogeneous fiber space over ${\mathrm{P}\mathrm{S}\mathrm{L}}_3\mathrm{/}K$ with fiber ${\mathbb{P}}^1$ for some subgroup $K$ isomorphic to the binary tetrahedral group ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_3)$.