#507510

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

446 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egi,c^2=ef,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, 414706);