We give an explicit description of the Loday assembly map on homotopy groups when restricted to a subgroup coming from the Atiyah-Hirzebruch spectral sequence. This proves and generalises a formula about the Loday assembly map on the first homotopy group that originally appeared in work of Waldhausen. Furthermore, we show that the Loday assembly map is injective on the second homotopy groups for a large class of integral group rings. Finally, we show that our methods can be used to compute the universal assembly map on homotopy.