Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\cdots,n$, there exists an isometry $f$ of $M$ such that $f(p_i)=p_{\sigma(i)}$ for all $i=1,2,\cdots,n$.
Question 1: Given $M$, for example, $M$ is the $m$-sphere, what is the largest integer $n$ such that there exists symmetric points $p_1,\cdots, p_n\in M$?
Question 2: Given $n$, how to determine all possible positions of symmetric $n$-points on $M$?
Question 3: Are there any backgrounds, motivations & related references for symmetric points on symmetric spaces?