Conjugacy classes of groups of prime order in $\mathrm{PGL}_{k+1}(\mathbb{C})$

Let $\mathbb{C}$ be the field of complex numbers. Let $k$ be natural number with $k \geq 2$ and let $p$ be a rational prime. In this paper we count the number of conjugacy classes of admissible cyclic subgroups of $\mathrm{PGL}_{k+1}(\mathbb{C})$ of order $p$, where with admissible we intend those finite subgroups that can be contained in the automorphism group of a set of points in $\mathbb{P}^k(\mathbb{C})$ in general position and of cardinality $n\geq k+3$. We also describe a kind of association between the conjugacy classes of these groups and show a beautiful relation connecting this type of association and the association between point sets.