TITLE: Is every distance-regular graph vertex-transitive?
QUESTION [7 upvotes]: Is every distance-regular graph vertex-transitive?

REPLY [4 votes]: Very small examples of distance-regular graphs that are not vertex-transitive are the Chang graphs on 28 vertices.  They are strongly regular but not vertex-transitive.  They are constructed from the very symmetric graph T(8).  The groups have sizes
http://en.wikipedia.org/wiki/Chang_graphs
http://www.win.tue.nl/~aeb/graphs/Chang.html 
384,360 and 96, which is not even divisibly by 28.
It is not true either that for large diameter, they have to be vertex-transitive.  In 2005, Koolen and van Dam constructed new distance-regular graphs of arbitrary diameter d, by twisting the very symmetric Grassmann graphs (hence the name "twisted Grassmann graph").  They have two orbits on vertices.