#509339

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,23)(13,30)(14,19)(16,21)(17,25)(18,24)(20,36)(22,34)(26,40)(27,29)(28,31)(32,38)(33,35)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,25)(10,17,19,15)(12,27,28,39)(13,30,18,24)(16,26,34,20)(21,36,22,40)(23,37,31,29)(32,35,33,38);(1,4,5,6)(2,3,7,8)(9,12)(10,18)(11,23)(13,19)(14,28)(15,35)(16,37)(17,38)(20,39)(21,32)(22,33)(24,36)(25,31)(26,27)(29,34)(30,40);(2,7)(3,8)(9,13,14,18)(10,12,19,28)(11,24,25,30)(15,27,17,39)(16,32,34,33)(20,35,26,38)(21,29,22,37)(23,36,31,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416535);