#490809

order	512
nilpotency class	4
isoclinism	isoclinic class of #490801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

491 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)(3,7)(6,8)(9,10)(11,20)(12,25)(13,19)(14,29)(15,30)(16,21)(17,23)(18,22)(24,37)(26,40)(27,36)(28,32)(31,33)(34,39)(35,38);(1,2,3,5,4,6,7,8)(9,11,14,23,16,24,13,15)(10,17,19,20,21,30,29,37)(12,22,27,38,28,39,26,31)(18,32,33,36,34,25,35,40);(9,12)(10,18)(11,22)(13,26)(14,27)(15,31)(16,28)(17,32)(19,33)(20,36)(21,34)(23,38)(24,39)(25,30)(29,35)(37,40);(2,6)(5,8)(9,13,16,14)(10,19,21,29)(11,15,24,23)(12,26,28,27)(17,20,30,37)(18,33,34,35)(22,31,39,38)(25,40,32,36)

Polycyclic presentation:

2^9|b^2=e,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=dgi,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398005);