#496929

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

469 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)(3,7)(9,10)(11,20,24,37)(12,17)(13,27)(14,25)(15,19)(16,23,34,40)(18,30)(21,29,33,39)(22,35,32,38)(26,28)(31,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,21,24,33)(12,25)(13,28)(14,17)(15,19)(16,22,34,32)(18,31)(20,29,37,39)(23,35,40,38)(26,27)(30,36);(2,5)(6,8)(9,11)(10,16)(12,22)(13,29)(14,32)(15,24)(17,33)(18,35)(19,34)(20,36)(21,25)(23,26)(27,37)(28,38)(30,40)(31,39);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,25)(11,22,24,32)(13,26,31,30)(16,33,34,21)(18,27,28,36)(20,35,37,38)(23,39,40,29)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

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, 404125);