For simplicity let's talk about $p$-localizations of spaces for a fixed prime $p$. Every space $X$ has a well-defined $p$-localization which can be constructed by the small object argument and which becomes a fibrant replacement in the $p$-local model structure on the category of spaces. It is well-known that nilpotent spaces have nice enough Postnikov towers and we can localize such spaces by taking the Postnikov tower, localizing step by step and putting it back together by taking the limit of the resulting tower of fibrations. My question is:

Is there an example of a non-nilpotent space $X$ whose $p$-localization we can explicitly describe?

I leave the meaning of "explicitly" ambiguous. I would be interested in any construction not using the small object argument.

Here's my stab at a possible example. For a group $G$ we define its lower central series by setting $G_0 = G$ and $G_{n + 1} = [G_n, G]$ and we can also continue transfinitely by setting $G_\beta = \bigcap_{\alpha < \beta} G_\alpha$ for limit ordinals $\beta$. The group $G$ is nilpotent if this construction terminates at the trivial subgroup at a finite stage. It is called *hypocentral* if it terminates at the trivial subgroup at some not necessarily finite stage. According to Wikipedia it is a result of Malcev that there are hypocentral groups with arbitrarily long lower central series.

If we start with a hypocentral group $G$ and convert its lower central series into a (transfinite) tower of fibrations (whose limit is a $K(G, 1)$), $p$-localize it step by step and take the limit of the resulting tower of fibrations, do we obtain the $p$-localization of $K(G, 1)$?