#507493

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
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 ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egi,c^2=f,d^2=ei,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,g^c=gi

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, 414689);