Inequalities between $\text{class}(G)$ and $\text{rank}(G/[G,G])$ for f.g. nilpotent $G$

Question

Let $G$ be a finitely generated nilpotent group of class $c$. Let $r=\text{rank}(G/[G,G])$ (i.e., the number of infinite cyclic summands of $G/[G,G]$).

Are there inequalities relating $c$ and $r$? (if so, are they tight?)

EDIT: There is a simple answer in the comments below.

Answer 1

Groups of maximal class are an interesting source of examples.

N. Blackburn has described among others the metabelian $p$-groups of maximal class, for $p > 3$. These are two-generated, and have arbitrarily large class. They can also be described handily with a modicum of number theory, see for instance the book

C.R. Leedham-Green and S. McKay, The structure of groups of prime power order.

The theory of basic commutators allows one to use free nilpotent groups as rather general examples.