#143101

order	512
nilpotency class	4
isoclinism	isoclinic class of #143098
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,24,30)(19,31,36,64)(20,26,40,56)(21,41,46,59)(22,29,23,28)(25,48,52,33)(27,44,60,45)(32,62,66,77)(34,57,35,63)(37,70,69,80)(38,72,43,74)(39,61,47,55)(42,50,68,51)(49,65,78,67)(53,75,79,71)(54,76,58,73);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,19,24,36)(18,25,30,52)(20,37,40,69)(21,42,46,68)(22,35,23,34)(26,53,56,79)(27,57,60,63)(28,51,29,50)(31,59,64,41)(32,47,66,39)(33,44,48,45)(38,67,43,65)(49,61,78,55)(54,62,58,77)(70,74,80,72)(71,76,75,73);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,20)(18,26)(19,32)(21,43)(22,47)(23,39)(24,40)(25,49)(27,58)(28,61)(29,55)(30,56)(31,62)(33,67)(34,69)(35,37)(36,66)(38,46)(41,74)(42,75)(44,73)(45,76)(48,65)(50,79)(51,53)(52,78)(54,60)(57,80)(59,72)(63,70)(64,77)(68,71)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50297);