Moduli Stacks of G-Curves in Homotopy Theory at h=p-1

We study the action on the deformation space of a formal group by the maximal finite subgroup G of its automorphisms, at the first height where the group has nontrivial p-torsion for odd p. We show given this group G there is a universal construction of a geometric model of the G-action via inverse Galois theory which generalizes the use of level structure to ramification data. We use configuration spaces to understand the model, and conclude that the Lubin-Tate action at h=p-1 is a subgroup of the symmetric group action on the configuration space of p+1 points on P¹.