#144100

order	512
nilpotency class	4
isoclinism	isoclinic class of #144085
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	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 70 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)(17,18,25,32)(19,33,39,64)(20,40,45,75)(21,29,48,61)(22,28,47,62)(23,31,24,30)(26,49,54,35)(27,55,59,42)(34,65,70,78)(36,52,46,53)(37,51,38,60)(41,57,76,58)(43,56,44,74)(50,68,79,69)(63,71,77,67)(66,73,72,80);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,26)(20,34)(21,46)(22,35)(23,38)(24,37)(25,39)(27,50)(28,60)(29,33)(30,53)(31,52)(32,54)(36,48)(40,73)(41,71)(42,66)(43,69)(44,68)(45,70)(47,49)(51,62)(55,72)(56,63)(57,65)(58,78)(59,79)(61,64)(67,76)(74,77)(75,80);(4,14)(6,15)(7,12)(10,13)(17,20,25,45)(18,27,32,59)(19,34,39,70)(21,42,48,55)(22,41,47,76)(23,44,24,43)(26,50,54,79)(28,40,62,75)(29,56,61,74)(30,58,31,57)(33,63,64,77)(35,71,49,67)(36,72,46,66)(37,69,38,68)(51,80,60,73)(52,65,53,78)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51296);