C₂×#484 #3090

order	128
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

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

230 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1822);