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C₈×#117 #20471

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×H(3,ℤ/4), #105261, C₈×#339, …
maximal normal subgroup of	C₈×H(3,ℤ/4), #105261, C₈×#339, …
maximal quotient of	C₈×#431, #183731, #183740, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13459);