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D₈×#99 #220280

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#99
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

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

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268288, #72268300, D₈×B(4,𝔽₂), …
maximal normal subgroup of	#72268288, #72268300, D₈×B(4,𝔽₂), …
maximal quotient of	#72313328, #99×#102, C₂×D₈×#99

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127476);