#494816

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

604 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)(9,10,12,20,15,21,14,18)(11,22,24,16,23,35,26,33)(13,28,27,36,31,17,30,19)(25,34,39,32,29,37,40,38);(2,5)(6,8)(9,11)(10,16,21,33)(12,24)(13,29)(14,26)(15,23)(17,32,28,38)(18,22,20,35)(19,37,36,34)(25,31)(27,40)(30,39);(1,3,4,7)(2,6,5,8)(10,17)(12,14)(13,27)(16,32)(18,19)(20,36)(21,28)(22,37)(24,26)(25,39)(29,40)(30,31)(33,38)(34,35);(11,23)(16,33)(22,35)(24,26)(25,29)(32,38)(34,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,e^2=i,f^2=h,g^2=i,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, 402012);