Spaces with finitely many non-trivial homotopy groups all of which are finite

It is well-known that the homotopy type of a connected CW-space can be faithfully represented by a simplicial group [1]. Under the assumption that the space has only finitely many non-trivial homotopy groups, all of which are finite, we show that this simplicial group may be chosen such that its group of n-simplices is finite for each n ⩾ 0.