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.



When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube
As it was mentioned in the comments, the pseudo-arc and the Hilbert cube have the fixed point property so they cannot be homeomorphic to a topological group.


On the other hand it was proved by G.S. Ungar in "*On all kinds of homogeneous spaces*" (TAMS, 1975), that any homogeneous compact metric space is homeomorphic to a coset space. In particular this is true for both the pseudo-arc and the Hilbert cube.


The fact that the pseudo-arc is a coset space was first proved by T.S. Wu in "*Each homogeneous nondegenerate chainable continuum is a coset space*" (PAMS, 1961). 


I don´t know who proved first that the Hilbert cube is a coset space, but it also follows from a theorem of L.F. Ford in "*Homeomorphism groups and coset spaces*" (TAMS, 1954), namely that any homogeneous strongly locally homogeneous Tychonoff space is a coset space.


