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C₄×#335 #8272

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#93268, #93270, #93406, …
maximal normal subgroup of	#93268, #93270, #93406, …
maximal quotient of	C₈×#335, #96888, #96899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1260);