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C₂×#1924 #21045

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator	C₂³	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂⁴	C₂²×C₄
prime radical	C₂⁵	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

699 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)(8,9)(11,12,15,17,14,13,16,18);(7,8,10,9)(11,13,15,18,14,12,16,17);(4,6)(7,9,10,8)(12,13)(17,18)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1924, #354330, #354332, …
maximal normal subgroup of	C₄×#1924, #354330, #354332, …
maximal quotient of	C₂×#9545, C₂×#9547, C₂×#9546, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14033);