Projective spaces and flag manifolds

Question

Is the complex projective space $\mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $\mathbb CP^2=S/H$?

Answer 1

More generally, for any sequence $d_1 < d_2 < \dots < d_k < n$, the space of all partial flags $$V_1 \subset V_2 \subset \dots \subset V_k \subset \Bbb C^n$$

with $\dim V_i = d_i$ has a transitive $SL_n(\Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.