ag.algebraic geometry – Conjugacy classes of plane k-jet group

Define $G(n, k)$ as a subgroup of $rm{Aut}(Bbb C(x_1, dots, x_n)/mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $Bbb C^n$). I’m interested in a map $G(2, k) to rm{Con(G(2, k))}$.

To be more precise, there are two subquestions.

  1. What is the space of conjugacy classes?

  2. Is there some reasonable combinatorial system describing fibers of that map?

I’d be happy to know anything about it in case of $Bbb C$ being replaced by finite field as well, not sure if it’s harder or easier. There are some results on enumerating conjugacy classes in triangular groups, for example, there’s only finite number of centralizers orbits for triangular matrices of size $<6$, and Kolchin thm may be helpful a bit.