#143161

order	512
nilpotency class	4
isoclinism	isoclinic class of #143154
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	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,24,30)(19,31,36,64)(20,37,41,75)(21,28,44,60)(22,27,45,59)(23,46,43,29)(25,49,53,33)(26,54,57,39)(32,65,69,78)(34,52,42,61)(35,51,47,58)(38,56,76,62)(40,55,48,74)(50,68,79,72)(63,70,77,67)(66,73,71,80);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19,24,36)(18,25,30,53)(20,32,41,69)(21,42,44,34)(22,33,45,49)(23,47,43,35)(26,50,57,79)(27,58,59,51)(28,31,60,64)(29,61,46,52)(37,73,75,80)(38,70,76,67)(39,66,54,71)(40,72,48,68)(55,63,74,77)(56,65,62,78);(6,15)(9,16)(17,20,24,41)(18,26,30,57)(19,32,36,69)(21,39,44,54)(22,38,45,76)(23,48,43,40)(25,50,53,79)(27,37,59,75)(28,55,60,74)(29,62,46,56)(31,63,64,77)(33,70,49,67)(34,71,42,66)(35,72,47,68)(51,80,58,73)(52,65,61,78)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,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, 50357);