Catalyzes the formation of the alpha-1,6-glucosidic linkages in glycogen by scission of a 1,4-alpha-linked oligosaccharide from growing alpha-1,4-glucan chains and the subsequent attachment of the oligosaccharide to the alpha-1,6 position. Transfers a segment of a (1->4)-alpha-D-glucan chain to a primary hydroxy group in a similar glucan chain. Glycan biosynthesis; glycogen biosynthesis. Monomer. Cells lacking this gene display maltose 1-phosphate (M1P) accumulation and trehalose sensitivity. These phenotypes are suppressed in mutants lacking both glgE and treS or both glgE and mak. Belongs to the glycosyl hydrolase 13 family. GlgB subfamily.