In this note, we study the cut locus of the free, step two Carnot groups G k with k generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Mya02, Mya06] and [MM16a], by exhibiting sets of cut points C k ⊂ G k which, for k ≥ 4, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension Θ(k 2) and semi-algebraic sets of codimension Θ(k), the sets C k are semi-algebraic and have codimension 2, yielding the best possible lower bound valid for all k on the size of the cut locus of G k. Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known that Abn0(G k) = Cut0(G k) \ Cut0(G k), k = 2, 3. For each k ≥ 4, instead, we show that the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of C k. The question whether C k coincides with the cut locus for k ≥4 remains open.