  Consider the diagonal action of the projective group $\PGL_3$ on $n$ copies
of ${\mathbb P}^2$. In addition, consider the action of the symmetric group
$\Sigma_n$ by permuting the copies. In this paper we find a set of generators
for the invariant field of the combined group $\Sigma_n \times \PGL_3$. As the
main application, we obtain a reconstruction principle for point configurations
in ${\mathbb P}^2$ from their sub-configurations of five points. Finally, we
address the question of how such reconstruction principles pass down to
subgroups.
