Quantum automorphism groups of small metric spaces

To any finite metric space $X$ we associate the universal Hopf \c^*-algebra $H$ coacting on $X$. We prove that spaces $X$ having at most 7 points fall into one of the following classes: (1) the coaction of $H$ is not transitive; (2) $H$ is the algebra of functions on the automorphism group of $X$; (3) $X$ is a simplex and $H$ corresponds to a Temperley-Lieb algebra; (4) $X$ is a product of simplexes and $H$ corresponds to a Fuss-Catalan algebra.