Lower central series and the center of $S/S^k$ Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. 
I don't know much about lower central series (and their connection to upper central series), but in my notes I have written $Z(S/S^2) \ge S^1/S^2$. Is this a general fact about lower central series, that $Z(S/S^k) \ge S^{k-1}/S^k$?
It looks familiar in terms of the upper central series, because here $Z(S/Z_k(S)) = Z_{k+1}(S)/Z_k(S)$. Is there a connection?

Let $xS^k \in S/S^k$ and $yS^k \in S^{k-1}/S^k$ with $x \in S$, $y \in S^{k-1}$. To show that $S^{k-1}/S^k \le Z(S/S^k)$, we have to show that $xS^k$ and $yS^k$ commute with each other or, in other words, that their commutator is the trivial element of $S/S^k$.
Now $[xS^k, yS^k] = [x,y]S^k$. But, since $x \in S$ and $y \in S^{k-1}$, $[x,y] \in [S,S^{k-1}] = S^k$. So $[x,y]S^k$ is indeed the trivial element of $S/S^k$.
