#496933

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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 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)(9,10,16,19,14,20,13,22)(11,23,24,33,12,17,26,35)(15,30,32,21,29,38,28,39)(18,31,37,27,36,40,34,25);(1,2)(3,6)(4,5)(7,8)(9,11,16,24,14,12,13,26)(10,17,19,35,20,23,22,33)(15,27,32,40,29,25,28,31)(18,21,37,38,36,39,34,30);(2,5)(6,8)(9,12,14,11)(10,18,20,36)(13,26,16,24)(15,31,29,40)(17,30,23,38)(19,34,22,37)(21,35,39,33)(25,28,27,32);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,33,23,35)(18,34,36,37)(21,38,39,30)(25,31,27,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=e,f^2=h,g^2=i,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, 404129);