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C₂×#1862 #20868

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

987 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);(3,4)(5,6)(7,8,10,9)(11,12)(13,16)(14,17)(15,18);(4,6)(7,9,10,8)(12,16)(13,14)(17,18);(12,17)(13,14)(16,18)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#99891, #99907, C₄×#1862, …
maximal normal subgroup of	#99891, #99907, C₄×#1862, …
maximal quotient of	C₂×#8843, C₂×#8847, #191888, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13856);