C₂³×D₈ #3574

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 2752512
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂³×D₈

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

988 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 14 points:

(1,2);(3,4);(5,6);(7,8,9,10,11,12,13,14);(8,14)(9,13)(10,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2306);