#507507

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 170 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,14,19,15,21,23,20)(11,16,24,34,13,31,32,18)(12,25,28,39,30,40,29,33)(17,26,35,27,36,38,37,22);(1,3,5,4)(2,6,8,7)(9,11,14,24,15,13,23,32)(10,16,19,34,21,31,20,18)(12,26,28,27,30,38,29,22)(17,25,35,39,36,40,37,33);(1,3,5,4)(2,7,8,6)(9,12,13,27,14,29,32,26,15,30,11,22,23,28,24,38)(10,17,18,25,20,35,31,39,21,36,34,40,19,37,16,33);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=ef,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,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, 414703);