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C₂×#336 #1727

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄	C₂³
socle, prime radical	C₂⁴	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

264 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);(3,4)(5,6)(11,12,13,15,14,16,17,18);(4,6)(7,8,9,10)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#92815, #92816, #92829, …
maximal subgroup of	#7016, #7020, #7025, …
maximal normal subgroup of	#7016, #7020, #7025, …
maximal quotient of	C₂×#1273, #7848, #7849, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 459);