Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus the harder question is, do they admit a structure of a quotient of a topological group divided (left or right) by a closed (not necessarily normal) subgroup?
One could also ask extra about other non-obvious homogenous topological spaces, and also about other topological algebraic structures different from the two mentioned above, i.e. from a topological group or its quotient by a closed subgroup.
Added: I was always curious (but didn't do much about it), if the pseudo-arc can be supplied with an interesting geometric structure, even if it is made up ad hoc for the pseudo-arc.