#504714

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	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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)(3,8)(4,6)(5,7)(9,10)(11,24)(12,27)(13,29)(14,21)(15,20)(16,22)(17,26)(18,23)(19,30)(25,33)(28,34)(31,40)(32,35)(36,39)(37,38);(1,3,5,4)(2,6,7,8)(9,11,15,26,17,20,24,10,18,21,27,22,16,12,14,23)(13,25,31,40,33,29,38,19,34,35,39,36,32,28,30,37);(1,4,5,3)(2,6,7,8)(9,12,14,22,16,21,27,10,18,20,24,26,17,11,15,23)(13,29,30,40,32,25,39,37,34,28,38,36,33,35,31,19);(2,7)(6,8)(9,13,15,31,17,33,24,38,18,34,27,39,16,32,14,30)(10,19,21,35,22,36,12,28,23,37,11,25,26,40,20,29)

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=f,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, 411910);