#507494

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=f,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=cg,d^b=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414690);