#143866

order	512
nilpotency class	4
isoclinism	isoclinic class of #143817
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 358 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) = 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)(19,25,34,50)(20,35,37,69)(21,41,45,57)(22,29,23,28)(26,51,53,38)(27,43,58,44)(31,60,61,79)(32,63,66,80)(33,59,46,49)(36,71,67,68)(39,56,40,70)(42,54,72,55)(47,73,75,62)(48,74,76,65)(52,64,77,78);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,25)(20,36)(21,32)(22,46)(23,33)(24,34)(26,52)(27,48)(28,59)(29,49)(30,50)(31,39)(35,68)(37,67)(38,64)(40,61)(41,63)(42,73)(43,74)(44,65)(45,66)(47,54)(51,78)(53,77)(55,75)(56,79)(57,80)(58,76)(60,70)(62,72)(69,71);(4,14)(6,15)(7,12)(10,13)(17,20)(18,26)(19,31)(21,42)(22,40)(23,39)(24,37)(25,47)(27,56)(28,55)(29,54)(30,53)(32,64)(33,67)(34,61)(35,41)(36,46)(38,44)(43,51)(45,72)(48,68)(49,77)(50,75)(52,59)(57,69)(58,70)(60,63)(62,65)(66,78)(71,76)(73,74)(79,80)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,d^c=dgi,e^c=eh,f^c=fi,h^c=hi

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, 51062);