#512295

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efi,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=ghi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419491);