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#41×#99 #6379503

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 16 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#41×B(4,𝔽₂), #99×AGL(1,ℤ/8), #72313062, …
maximal normal subgroup of	#41×B(4,𝔽₂), #99×AGL(1,ℤ/8), #72313062, …
maximal quotient of	C₄×D₄×#99

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286699);