  The stratified structure of the configuration space $\mb G^N = G \times ...
\times G$ reduced with respect to the action of $G$ by inner automorphisms is
investigated for $G = SU(3) .$ This is a finite dimensional model coming from
lattice QCD. First, the stratification is characterized algebraically, for
arbitrary $N$. Next, the full algebra of invariants is discussed for the cases
$N = 1$ and $N =2 .$ Finally, for $N = 1$ and $N =2 ,$ the stratified structure
is investigated in some detail, both in terms of invariants and relations and
in more geometric terms. Moreover, the strata are characterized explicitly
using local cross sections.
