The Foulkes conjecture states that the multiplicities in the plethysm Syma(SymbV) are at most as large as the multiplicities in the plethysm Symb(SymaV) 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.