[GAP Forum] Transfinitely nilpotent groups

[Nagy Gábor]

Dear Forum,

Can somebody gime me a construction of a transfinitely upper nilpotent 
group of class bigger than omega?

That is, I define the upper central series of G by Z_0(G)=1, 
Z_{k+1}(G)/Z_k(G)=Z(G/Z_k(G)) and Z_\kappa(G)=U_{k<\kappa} Z_k(G) for 
limit ordinals.

I have Z_\kappa(G)=G for some ordinal \kappa. Are there examples where 
\kappa>\omega? In particular, what about \kappa=\omega+1?

Thanks in advance, bye,

Gabor