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C₂×#331 #1538

order	128
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
prime radical	C₂×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#7485, C₄×#331, #8231, …
maximal normal subgroup of	#7485, C₄×#331, #8231, …
maximal quotient of	C₂×#1270, #7569, #7640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 270);