#491798

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	#33840	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

467 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^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, 398994);