Nilpotent not implies UL-equivalent

Statement
A nilpotent group need not be UL-equivalent: its fact about::upper central series and fact about::lower central series need not coincide.

Conditions strong enough to imply UL-equivalent
See UL-equivalent group.

Lower central series versus upper central series
Here are some contrasts between facts that are true for lower central series and upper central series. Each of these pairs of facts gives further examples of nilpotent groups that are not UL-equivalent.





Examples
Below are links to lists of examples:



A direct product of groups of different nilpotence classes
Suppose $$G,H$$ are groups of nilpotency classes $$c,d$$ respectively, with $$c < d$$. Then, the direct product $$G \times H$$ is a nilpotent group.

The lower central series and upper central series are both coordinate-wise. Hence, if $$\gamma_i(G)$$ denotes the $$i^{th}$$ term of the lower central series with $$\gamma_1(G) = G$$, then:

$$\gamma_i(G \times H) = \gamma_i(G) \times \gamma_i(H)$$

Thus, we have:

$$\gamma_k(G \times H) = \{ e \} \times \gamma_k(H) \ \forall \ k \ge c+1$$

On the other hand, we have:

$$Z^{d+1-k}(G \times H) = Z^{d+1-k}(G) \times Z^{d+1-k}(H)$$

In particular, if $$c+1 \le k < d + 1$$, then $$\gamma_k(G \times H)$$ has trivial $$G$$-projection, but $$Z^{d+1-k}(G \times H)$$ does not.

Particular examples
The smallest concrete examples of the direct product construction are given below:

Some examples that are not part of this general construction are nontrivial semidirect product of Z4 and Z4, central product of D8 and Z4, and holomorph of Z8.