The block-transitive,point-imprimitive 2-(729, 8, 1) designs

The block-transitive point-imprimitive 2-(729,8,1) designs are classified. They all have full automorphism group of order 729.13 which is an extension of a groupN of order 729, acting regularly on points, by a group of order 13. There are, up to isomorphism, 27 designs withN elementary abelian, 13 designs withN=Z ₉ ³ and 427 designs withN the relatively free 3-generator, exponent 3, nilpotency class 2 group, a total of 467 designs. This classification completes the classification of block-transitive, point-imprimitive 2-(, k, 1) designs satisfying , which is the Delandtsheer-Doyen upper bound for the number of points of such designs. The only examples of block-transitive, point-imprimitive 2-(, k, 1) designs with are the 2-(729, 8, 1) designs constructed in this paper.The first three authors acknowledge the support of an Australian European Awards Program scholarship, a Deutsche Akademische Austauschdienst scholarship, and an Australian Research Council Research Fellowship, respectivelyThe authors wish to thank Brendan McKay for his independent verification of the non-isomorphism of the classes of designs found, and of their automorphism groups, using different, nauty techniques [6].