#507454

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 414650);