TITLE: Smallest strongly regular graph whose automorphism group is not vertex transitive?
QUESTION [12 upvotes]: I'm looking for a small strongly regular graph whose automorphism group is not vertex-transitive.
This answer to a different question shows that the Chang graphs on 28 vertices are such graphs.  Is there an example on less vertices?
Thanks for all replies.

REPLY [13 votes]: There is no smaller example.
Various places, including Andries Brouwer's list of parameters and existence for small SRGS (http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html) show that there are only a handful of parameter sets to check.
Those with fewer than 25 vertices can almost be checked by hand as they fall into a few families (Paley graphs) or are well-known individual graphs (e.g Clebsch graph). 
For the parameter set  $(25, 12, 5, 6)$ there are exactly $15$ graphs and they have automorphism groups of orders $1$ (twice), $2$ (four times), $3$ (twice), $6$ (four times), $72$ (twice) and $600$.