#491831

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	#3287	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 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)(11,18)(12,23)(13,19)(14,28)(15,16)(17,26)(20,30)(21,37)(22,35)(24,36)(25,39)(27,32)(29,34)(31,40)(33,38);(1,2,4,7)(3,6,8,5)(9,11,13,15)(10,16,19,18)(12,21,25,24)(14,22,27,30)(17,31,33,34)(20,32,35,28)(23,36,39,37)(26,29,38,40);(9,12,13,25)(10,17,19,33)(11,21,15,24)(14,29,27,40)(16,31,18,34)(20,36,35,37)(22,38,30,26)(23,32,39,28);(2,6)(3,8)(5,7)(9,11,13,15)(10,18,19,16)(12,24,25,21)(14,22,27,30)(17,31,33,34)(20,28,35,32)(23,36,39,37)(26,40,38,29)

Polycyclic presentation:

2^9|b^2=gi,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, 399027);