C₂×#474 #3070

order	128
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 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);(3,4,6,8)(5,9,10,7)(11,12)(13,15)(14,17)(16,18);(3,5,6,10)(4,7,8,9);(12,15)(13,18)(16,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1802);