The Wells graph, sometimes also called the Armanios-Wells graph, is a quintic graph on 32 nodes and 80 edges that is the unique distance-regular graph with intersection array. It is also distance-transitive. It is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266). It is illustrated above in a number of non-LCF embeddings.
The Wells graph possesses at least 2 order-8 LCF embeddings, 9 of order 4, and 3 bilaterally symmetric of order 2, as illustrated above.
Armanios, C. "Symmetric Graphs and Their Automorphism Groups." Ph.D. thesis. Perth, Australia: University of Western Australia, 1981.Armanios, C. "A New 5-Valent Distance Transitive Graph." Ars Combin.19A, 77-85, 1985.Brouwer, A. E. "Armanios-Wells Graph." http://www.win.tue.nl/~aeb/drg/graphs/Wells.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Covers of Cubes and Folded Cubes--The Wells Graph." §9.2E in Distance Regular Graphs. New York: Springer-Verlag, pp. 266-267, 1989.DistanceRegular.org. "Armanios-Wells Graph." https://www.math.mun.ca/distanceregular/graphs//armanios-wells.html.van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin.15, 189-202, 2003.Wells, A. L. "Regular Generalized Switching Classes and Related Topics." Ph.D. thesis. Oxford, England: University of Oxford, 1983.
. It is also distance-transitive. It is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266). It is illustrated above in a number of non-LCF embeddings. 
(van Dam and Haemers 2003). 