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C₂×#332 #1539

order	128
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

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

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#332, #8233, #8236, …
maximal normal subgroup of	C₄×#332, #8233, #8236, …
maximal quotient of	C₂×#1271, #7570, #7641, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 271);