BLOCK-TRANSITIVE, POINT-IMPRIMITIVE DESIGNS WITH LAMBDA = 1

Let D be a 2-(v, k, 1) design with a group G of automorphisms which is transitive on the blocks of D and transitive but imprimitive on the p oints of D. Delandtsheer and Doyen (1989) proved that v is bounded abo ve by (k - 2)2(k + 1)2/4. Carrying on from the work of Cameron and Pra eger (1989), we show that if v is equal to this upper bound then v = 7 29 and k = 8. Further work of Nickel et al. (1992) has shown that, up to an isomorphism, there are 467 block-transitive, point-imprimitive 2 -(729, 8, 1) designs.