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C₂×#1880 #20923

order	256
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

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

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#99898, #99914, C₄×#1880, …
maximal normal subgroup of	#99898, #99914, C₄×#1880, …
maximal quotient of	C₂×#8948, C₂×#8952, #193575, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13911);