  We study noncommutative differential structures on the group of permutations
$S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior
algebra $\Lambda_N$ which is a super analogue of the Fomin-Kirillov algebra
$\CE_N$ for Schubert calculus on the cohomology of the $GL_N$ flag variety.
Noncommutative de Rahm cohomology and moduli of flat connections are computed
for $N<6$. We find that flat connections of submaximal cardinality form a
natural representation associated to each conjugacy class, often irreducible,
and are analogues of the Dunkl elements in $\CE_N$. We also construct
$\Lambda_N$ and $\CE_N$ as braided groups in the category of $S_N$-crossed
modules, giving a new approach to the latter that makes sense for all flag
varieties.
