order	128
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

Q₈×Q₁₆

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

168 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2114);