#491803

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
prime radical	#19908	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

454 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)(11,20,24,38)(12,17,14,30)(13,26,29,37)(16,31,33,22)(18,25,36,28)(21,35,23,40)(27,34,39,32);(1,2,4,7)(3,6,8,5);(9,11,15,24)(10,16,19,33)(12,23,14,21)(13,27,29,39)(17,34,30,32)(18,35,36,40)(20,37,38,26)(22,25,31,28);(2,6)(3,8)(5,7)(9,12,15,14)(10,17,19,30)(11,21,24,23)(13,25,29,28)(16,32,33,34)(18,26,36,37)(20,35,38,40)(22,39,31,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 398999);