#504290

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(11,15,25,29)(12,26,20,14)(13,24,33,40)(16,21,18,22)(19,35,38,30)(27,32,28,34)(31,37,39,36);(2,5)(6,7)(9,11,15,23,16,12,14,21,17,25,29,10,18,20,26,22)(13,24,31,38,32,27,30,36,33,40,39,19,34,28,35,37);(9,12,14,10,18,11,15,21,17,20,26,23,16,25,29,22)(13,28,30,38,34,40,31,37,33,27,35,19,32,24,39,36);(1,3,4,8)(2,6,5,7)(9,13,29,39,16,32,26,35,17,33,15,31,18,34,14,30)(10,19,12,27,22,37,25,40,23,38,20,28,21,36,11,24)

Polycyclic presentation:

2^9|a^2=ef,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, 411486);