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C₂×#1371 #12006

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2);(3,4,5,7,6,8,9,10)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23);(11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94021, #94026, #94044, …
maximal normal subgroup of	#94021, #94026, #94044, …
maximal quotient of	C₂×#7100, #122932, #122936, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4994);