#143449

order	512
nilpotency class	4
isoclinism	isoclinic class of #143446
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50645);