#143862

order	512
nilpotency class	4
isoclinism	isoclinic class of #143811
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	C₂³	#449
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

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

normal subgroups only

This group has 352 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)(6,10)(7,14)(11,16)(13,15)(17,18,24,30)(19,25,34,51)(20,35,40,71)(21,41,46,59)(22,29,23,28)(26,52,56,37)(27,44,60,45)(31,62,64,80)(32,65,42,74)(33,61,47,50)(36,69,68,78)(38,58,39,70)(43,54,73,55)(48,76,77,63)(49,75,57,67)(53,66,79,72);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,25)(20,36)(21,42)(22,47)(23,33)(24,34)(26,53)(27,57)(28,61)(29,50)(30,51)(31,38)(32,46)(35,69)(37,72)(39,64)(40,68)(41,74)(43,63)(44,67)(45,75)(48,54)(49,60)(52,66)(55,77)(56,79)(58,62)(59,65)(70,80)(71,78)(73,76);(4,14)(6,15)(7,12)(10,13)(17,20,24,40)(18,26,30,56)(19,31,34,64)(21,43,46,73)(22,39,23,38)(25,48,51,77)(27,58,60,70)(28,55,29,54)(32,66,42,72)(33,68,47,36)(35,59,71,41)(37,44,52,45)(49,78,57,69)(50,79,61,53)(62,74,80,65)(63,75,76,67)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,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,e^b=ei,f^b=fi,d^c=dg,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, 51058);