It is well known that the symmetric groupSntogether with one idempotent of rankn- 1 on a finiten-element setNserves as a set of generators for the semigroupTnof all the total transformations onN. It is also well known that the singular part SingnofTncan be generated by a set of idempotents of rankn- 1. The purpose of this paper is to begin an investigation of the way in which Singnand its subsemigroups can be generated by the conjugates of a subset of elements ofTnby a subgroup ofSn. We look for the smallest subset of elements ofTnthat will serve and, correspondingly, for a characterization of those subgroups ofSnthat will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rankn- 1 underGsuffice to generate Singnif and only ifGis what we define to be a 2-block transitive subgroup ofSn.