Products of commutators and the order of a finite group

Let G be a finite nonabelian group. Define λ(G) to be the smallest positive integer such that every element of G′ is expressible as a product of λ(G) commutators. We show |G′|⩾(λ(G)+1)!(λ(G)−1)!. We use this result to prove a conjecture of V.G. Bardakov in the Kourovka notebook that , with the bound obtained only at the group S3.