#507603

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414799);