#143521

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
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

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

normal subgroups only

This group has 350 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,24,30)(19,31,34,61)(20,26,40,55)(21,41,45,57)(22,29,23,28)(25,47,49,33)(27,43,58,44)(32,63,65,60)(35,56,36,62)(37,68,67,80)(38,70,73,79)(39,59,46,54)(42,50,66,51)(48,64,75,71)(52,77,76,69)(53,74,78,72);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,24,34)(18,25,30,49)(20,37,40,67)(21,42,45,66)(22,36,23,35)(26,52,55,76)(27,56,58,62)(28,51,29,50)(31,57,61,41)(32,46,65,39)(33,43,47,44)(38,71,73,64)(48,59,75,54)(53,63,78,60)(68,79,80,70)(69,74,77,72);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,20)(18,26)(19,32)(21,38)(22,46)(23,39)(24,40)(25,48)(27,53)(28,59)(29,54)(30,55)(31,60)(33,64)(34,65)(35,67)(36,37)(41,70)(42,69)(43,74)(44,72)(45,73)(47,71)(49,75)(50,76)(51,52)(56,68)(57,79)(58,78)(61,63)(62,80)(66,77)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50717);