Suppose $G$ is a group and $S \subset G$ is its finite subset. Let’s define the commuting graph of $G$ in respect to $S$ as $Comm(G, S)$ - a graph, where the vertices are the elements of $S$, and they are connected with edges iff they commute.
Let’s define the commuting graph universality of $G$ as $CGU(G)$ - the maximal $k \in \mathbb{N}$ such that any graph with $k$ vertices is a commuting graph of $G$ for some $S$.
What is the minimal possible order of $G$, such that $CGU(G) = n$?
The constructions from the questions «Commuting graphs of groups» and «What is the minimal possible size of an $n$-universal graph?» provide us with such a group of order $2^{(1+o(1)2^{\frac{n-1}{2}}}$.
However, a much smaller example (of order $(2n(n-1))!$) is possible. This is $S_{2n(n-1)}$.
Indeed, suppose $A = \{a_{ij}\}_{1 \leq i,j \leq n} \cup \{b_{ij}\}_{1 \leq i,j \leq n}$. For a graph $\Gamma(V, E)$, where $V = \{1, ... , n\}$, we can take $S_\Gamma \subset Sym(A)$, defined by the following formula $S = \{(a_{1i}b_{1i}a_{2i}b_{2i}…a_{ni}b_{ni})\Pi_{(i, j) \in E}(a_{ji}b_{ji})|i \in V\}$.
However, I do not know, whether there is a room for further improvement…