#146107

order	512
nilpotency class	4
isoclinism	isoclinic class of #146104
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#488
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

189 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=d,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=egi,f^c=fgh,h^c=hi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53303);