C₂³×Q₁₆ #3576

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 2752512
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂³×Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₄
commutator, Frattini, cosocle	C₄	C₂⁵
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2308);