#502022

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33611	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,22)(12,26)(13,31)(14,20)(15,35)(16,21)(17,25)(18,28)(19,32)(23,39)(24,38)(27,29)(30,34)(33,36)(37,40);(1,3,4,7)(2,5,6,8)(9,11,23,38,15,26,13,25,16,27,32,40,14,34,33,18)(10,17,36,12,20,24,19,22,21,28,31,30,35,37,39,29);(2,6)(5,8)(9,10)(11,24)(12,18)(13,19)(14,35)(15,20)(16,21)(17,34)(22,38)(23,36)(25,30)(26,28)(27,37)(29,40)(31,32)(33,39);(9,12,23,24,15,22,13,28,16,30,32,37,14,29,33,17)(10,18,36,11,20,38,19,26,21,25,31,27,35,40,39,34)

Polycyclic presentation:

2^9|b^2=efhi,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409218);