If $G$ is a finite $p$-group and the commutator subgroup $[G',\gamma_i(G)]\not=1$, can we ensure that the index $|[G',\gamma_i(G)]:[G',\gamma_{i+1}(G)]|\geq p$?
Here $G'$ is the derived subgroup, and $\gamma_i(G)$ is the $i$-th term of the lower central series of $G$.
Note that since $G$ is a finite $p$-grou, $G$ is nilpotent therefore if $\gamma_i(G)$ is not trivial, then $|\gamma_i(G):\gamma_{i+1}(G)|\geq p$. So it is clear that for a sufficiently big $j$ be can always find that $|[G',\gamma_i(G)]:[G',\gamma_{i+j}(G)]|\geq p$.
So I think it might be true. But I have no good reason to support this. I have tried to find a counterexample and I have failed. Does anyone have an idea on how to prove or disprove this claim?
It's not true. In the GAP small groups library there are $4$ groups of order $128$ (with ids $134, 135, 136, 137$) of nilpotency class $5$ and derived length $3$. Each of them satisfies $G'' = [G', G'] = [G', \gamma_3(G)] \cong C_2$.