#512354

order	512
nilpotency class	4
isoclinism	isoclinic class of #511998
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	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

517 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)(9,10,15,20)(11,21,24,19)(12,25)(13,30,22,18)(14,16,32,33)(17,23)(26,37)(27,36)(28,38)(29,40)(31,35)(34,39);(1,3,4,7)(2,5,6,8)(9,11,22,32,15,24,13,14)(10,16,30,19,20,33,18,21)(12,23,28,39,29,27,26,31)(17,25,35,37,36,40,34,38);(2,6)(5,8);(9,12,13,26,15,29,22,28)(10,17,18,34,20,36,30,35)(11,23,14,31,24,27,32,39)(16,25,21,38,33,40,19,37)

Polycyclic presentation:

2^9|a^2=eh,b^2=efi,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=gh,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, 419550);