TITLE: Nilpotent group with ascending and descending central series different?
QUESTION [6 upvotes]: This may turn out to be a bit embarassing, but here it is. It is probably not true that the ascending and descending central series (*) of a nilpotent group have the same terms. (Or at least one of MacLane-Birkhoff, Rotman and Jacobson would have mentioned it.) However, I have been unable to find an example where they are different. I thought I had a sketch of proof that they are always equal, but there is a gap, of the kind where you feel it is not patchable.
I've proved it for a few nilpotent groups (dihedral of the square, any group of order p^3, the Heisenberg groups of dimension 3 and 4 over any ring -- I think the argument extends to any dimension), and checked a few more exotic examples in the excellent Group Properties Wiki. So,

What is the simplest (preferably finite) nilpotent group such that its a.c.s. and d.c.s. are different?

and

Do the a.c.s. and d.c.s. coincide in some interesting general case?

(*) For completeness, the ascending central series of a group G is defined by Z_0 = 1, Z_{i+1} = the pullback of Z(G/Z_i(G)) along the projection, and the descending central series by G_0 = G, G_{i+1} = [G,G_i]. The group G is nilpotent iff ever Z_m = G or G_m = 1. It turns out that if such m exists it is the same for both.

REPLY [3 votes]: Not only can the two central series be different, there are many qualitative properties that the descending central series (also called the lower central series) satisfies that the ascending central series (also called the upper central series) usually does not. For instance, the lower central series is strongly central (as pointed out by Tom Church in his comment, though he didn't use that word) but the upper central series (turned around) need not be. At least half the members of the lower central series are abelian, but this need not be true of the upper central series. All the members of the lower central series are verbal subgroups, but the members of the upper central series need not even be fully invariant.
For more related information, see this page:
http://groupprops.subwiki.org/wiki/Nilpotent_not_implies_UL-equivalent
[DISCLOSURE: I wrote the material on that page and in most of the linked-to pages]