Let $G$ be an abelian permutation group acting on a set $\Omega$ such that $|G| = n$ and all its elements are of order at most 2. Let $g_1, \ldots, g_k \in G$ be distinct elements and suppose that $G = \langle g_1, \ldots, g_k\rangle$.

Question: Is $k$ necessarily small (logarithmic) in $n$ or can it be large (polynomial)?