#504009

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

563 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)(12,29)(13,35)(14,23)(15,22)(16,24)(17,32)(18,25)(19,21)(20,34)(27,37)(28,36)(30,39)(31,40)(33,38);(2,5)(6,7)(9,11,15,27,17,28,31,33,18,12,30,34,16,35,14,21)(10,19,23,13,24,20,39,29,25,38,40,36,32,37,22,26);(9,12,31,21,16,28,15,34,18,11,14,33,17,35,30,27)(10,20,39,36,32,19,23,29,25,37,22,13,24,38,40,26);(1,3,4,8)(2,6,5,7)(9,13,15,20,17,29,31,38,18,36,30,37,16,26,14,19)(10,21,23,11,24,27,39,28,25,33,40,12,32,34,22,35)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=eh,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,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411205);