#494813

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402009);