#507617

order	512
nilpotency class	4
isoclinism	isoclinic class of #500887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,c^2=g,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414813);