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C₈×#442 #183868

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×AGL(1,ℤ/16), C₂×C₈×#442, SD₁₆×#442, …
maximal normal subgroup of	C₈×AGL(1,ℤ/16), C₂×C₈×#442, SD₁₆×#442, …
maximal quotient of	C₈×#1580, C₈×AGL(1,ℤ/16), C₂×C₈×#442, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91064);