#143665

order	512
nilpotency class	4
isoclinism	isoclinic class of #143663
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

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

normal subgroups only

This group has 355 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fgh,g^c=gi,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, 50861);