  The Foulkes conjecture states that the multiplicities in the plethysm
Sym^a(Sym^b V) are at most as large as the multiplicities in the plethysm
Sym^b(Sym^a V) for all a <= b. This conjecture has been known to be true for a
<= 4. The main result of this paper is its verification for a = 5. This is
achieved by performing a combinatorial calculation on a computer and using a
propagation theorem of Tom McKay from 2008.
  Moreover, we obtain a complete representation theoretic decomposition of the
vanishing ideal of the 5th Chow variety in degree 5, we show that there are no
degree 5 equations for the 6th Chow variety, and we also find some
representation theoretic degree 6 equations for the 6th Chow variety.
