#504008

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	#33722	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 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)(3,7)(4,5)(6,8)(9,10)(11,26)(12,31)(13,35)(14,37)(15,38)(16,24)(17,28)(18,25)(19,21)(20,34)(22,32)(23,27)(29,39)(30,36)(33,40);(2,5)(6,7)(9,11,15,29,17,30,14,21,18,12,32,34,16,35,27,33)(10,19,23,36,24,39,22,26,25,40,37,13,28,20,38,31);(9,12,14,33,16,30,15,34,18,11,27,21,17,35,32,29)(10,20,22,36,28,40,23,31,25,39,38,13,24,19,37,26);(1,3,4,8)(2,6,5,7)(9,13,15,20,17,31,14,19,18,36,32,39,16,26,27,40)(10,21,23,12,24,34,22,35,25,33,37,11,28,29,38,30)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=ehi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,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, 411204);