There is much activity around the study of highly connected covers of Lie groups. Looking at the resulting homotopy groups, one starts with a sequence of period 8 (Bott periodicity), replaces some starting terms with zeroes, and leaves the remaining tail intact. Current names for the results seem to be string, fivebrane, etc. groups.

On the other hand, in view of things like tmf and higher chromatic phenomena it is natural to replace with zeroes an infinite subsequence of some higher period (24 or 576 or...). This would correspond to taking homotopy fibers of maps from the group not to an Eilenberg-MacLane space but rather to some space which itself has periodic homotopy groups - either to an infinite product of Eilenberg-MacLane spaces (killing a periodic sequence of characteristic classes), or to something more subtle having nontrivial k-invariants, I have no idea what.

Have such constructions been carried out? What are the resulting groups/H-spaces and their classifying spaces?