#491763

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
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	#19960	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

462 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)(11,20)(12,18)(13,16)(14,27)(15,30)(17,23)(19,29)(21,35)(22,36)(24,40)(25,32)(26,37)(28,34)(31,39)(33,38);(1,2)(3,6)(4,7)(5,8)(10,16)(17,31)(18,30)(19,32)(20,36)(27,37)(33,34)(35,40);(9,11,13,22)(10,17,16,31)(12,24,15,21)(14,28,26,38)(18,34,30,33)(19,35,32,40)(20,37,36,27)(23,25,39,29);(2,6)(3,8)(5,7)(9,12,13,15)(10,18,16,30)(11,21,22,24)(14,25,26,29)(17,33,31,34)(19,27,32,37)(20,35,36,40)(23,38,39,28)

Polycyclic presentation:

2^9|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=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

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, 398959);