Let $\Pi$ denote the space of all permutations of $\{1,\dots,n\}$, and let $d(\cdot,\cdot)$ be a metric on $\Pi$. Suppose I am given a large integer $K$ and I have to select $K$ permutations $\pi_1,\dots,\pi_K$ so as to minimize the maximum distance between any permutation $\pi\in\Pi$ and one of the selected $K$ permutations, that is, I want to solve the following optimization problem: $$\mathrm{minimize}_{\pi_1,\dots,\pi_K\in \Pi} \max_{\pi\in\Pi} \min_k d(\pi,\pi_k) $$In other words, I'm looking for a set of $K$ permutations that is as "spread out" in $\Pi$ as possible.

My question: are there any well-known metrics $d(\cdot,\cdot)$ that allow me to bound the distance above in terms of $K$? In other words, are there any statements of the form "if $K=(*)$ and metric $(**)$ is used, then it is possible to select $K$ centers such that every permutation $\pi$ is at most $(***)$ distance away from its nearest center"? Obviously, $K$ will need to be huge for this to make sense, so I'm particularly interested in the case where $K=(aN)!$ for $a$ close to $1$, for example.