#142929

order	512
nilpotency class	4
isoclinism	isoclinic class of #142831
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	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,24,30)(19,31,34,62)(20,26,41,58)(21,42,44,28)(22,27,45,61)(23,46,43,29)(25,49,51,33)(32,54,69,66)(35,64,71,53)(36,52,47,59)(37,73,74,50)(38,60,76,56)(39,55,67,63)(40,65,48,57)(68,80,75,78)(70,79,72,77);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,24,34)(18,25,30,51)(20,37,41,74)(21,35,44,71)(22,33,45,49)(23,47,43,36)(26,54,58,66)(27,59,61,52)(28,62,42,31)(29,64,46,53)(32,67,69,39)(38,70,76,72)(40,68,48,75)(50,60,73,56)(55,80,63,78)(57,79,65,77);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,20,24,41)(18,26,30,58)(19,32,34,69)(21,39,44,67)(22,38,45,76)(23,48,43,40)(25,50,51,73)(27,60,61,56)(28,63,42,55)(29,65,46,57)(31,66,62,54)(33,37,49,74)(35,68,71,75)(36,72,47,70)(52,79,59,77)(53,80,64,78)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50125);