#143195

order	512
nilpotency class	4
isoclinism	isoclinic class of #143188
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	#459
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 345 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,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, 50391);