#509826

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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)(4,6)(5,8)(9,10,17,23,35,29,36,37,16,12,14,20,15,21,38,34)(11,18,19,13,25,22,39,31,27,28,24,33,26,30,40,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,18)(12,28)(13,23)(14,24)(15,26)(16,27)(17,19)(20,33)(21,30)(22,29)(25,35)(31,37)(32,34)(36,39)(38,40);(9,12,16,10)(11,22,27,30)(13,24,33,19)(14,34,17,37)(15,21,35,29)(18,26,28,25)(20,36,23,38)(31,39,32,40);(2,6)(3,8)(9,13)(10,19)(11,23)(12,24)(14,28)(15,31)(16,33)(17,18)(20,27)(21,39)(22,38)(25,34)(26,37)(29,40)(30,36)(32,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417022);