The commutator x-1y-1xy of x and y is denoted by [x,y]. It satisfies the identity xy = yx [x,y].	The iterated commutator [x, n+1y] is defined recursively by [x, 1y] = [x,y] [x, n+1y] = [[x, ny], y].
G is called an Engel group if for all elements g and h in G there is a positive integer n (which depends on g an h) such that [g, nh] = 1 in G.	G is called an n-Engel group (or Engel-n group) if there is a positive integer n such that [g, nh] = 1 in G for all elements g and h in G.
The free nilpotent d-generator Engel-n group is denoted by E(d,n). Since [x, ny] is an element of the (n+1)th term of the lower central series, the class-n quotient of E(d,n) is the free nilpotent group of rank d and class n.

	class	Hirsch length	torsion subgroup	lower central series
E(2,4)	6	11	1	Z2, Z, Z2, Z3, C2×Z2, Z
E(3,4)	9	88	C544× C105× C304× C604	Z3, Z3, Z8, Z18, Z24× C24× C105× C30, Z26× C59× C109, Z6× C523× C10× C303, C53× C303, C3
E(2,5)	9	23	C38× C303× C1802	Z2, Z, Z2, Z3, Z6, C2×C6× Z4, C62× C182× Z4, C2×C303× Z, C34× C152
E(2,6)	12	70	C75× C1415× C8410× C1683× C8402× C2520× C126003× C321564600 2	Z2, Z, Z2, Z3, Z6, Z9, C2× C62× Z12, C23× C422× C843× Z13, C29× C426× C1262× Z14, C213× C143× C428× C1050× C63002× Z8, C27× C1410× C2102× C53594100 2, C22× C10
E(2,7)	> 11	> 149