#491743

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
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	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=de,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

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, 398939);