Alpha-transglucosylase that specifically transfers single glucosyl units from alpha(1->4)-glucans to the non-reducing terminal 4-OH of glucose and alpha(1->4)- and alpha(1->6)-linked glucosyl residues. Acts on amylose, amylopectin, glycogen and maltooligosaccharides, with the highest activity with maltotriose as a donor, and also accepts maltose. Does not act as a hydrolase: weak hydrolysis activity is only observed on the disaccharide maltose. Transfers the non-reducing terminal alpha-D-glucose residue from a (1->4)-alpha-D-glucan to the 4-position of a free glucose or of a glucosyl residue at the non-reducing terminus of a (1->4)-alpha-D-glucan, thus bringing about the rearrangement of oligosaccharides. kcat is 341 sec(-1) for maltose. kcat is 239 sec(-1) for maltotriose. kcat is 123 sec(-1) for maltotetraose. kcat is 181 sec(-1) for maltopentaose. Belongs to the glycosyl hydrolase 31 family.