#507455

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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,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, 414651);