#504292

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

496 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:

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

Polycyclic presentation:

2^9|a^2=ef,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,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, 411488);