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For a prime number p and a free profinite group S, let S (n,p) be the nth term of its lower p-central filtration, and S [n,p] the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group H 2 (S [n,p] , Z/p), which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of S (n,p) /S (n+1,p) . We prove that the cohomology group satisfiesdoi:10.4171/dm/584 fatcat:675ikjw32bagvbo4ovzgdpscle