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C₂×#361 #2146

order	128
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	Aut(G) order 2048
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	#102
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

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

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#361, #12647, #12736, …
maximal normal subgroup of	C₄×#361, #12647, #12736, …
maximal quotient of	C₂×#1333, #11599, #11652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 878);