Notation: $\mathbb{P}^2$ denotes complex projective plane.
We have an action $$SL_3 \times \mathbb{P}^2 \to \mathbb{P}^2, \ (A,[v])\mapsto [Av]$$ with kernel $\mathbb{C}^*.Id\cap SL_3 \cong C_3$ where $C_3$ is cyclic group of order $3$ generated by a third root of unity. $SL_3 / C_3 = PSL_3$ acts effectively on $\mathbb{P}^2$. Hence, we have an injective homomorphism $$ \phi: PSL_3 \hookrightarrow Aut(\mathbb{P}^2).$$
Is it true that $PSL_3 \cong Aut(\mathbb{P}^2)$ ? How to prove this? I know that $Aut(\mathbb{P}^2) \cong PGL_3$? But I am not sure how does this help?