#142977

order	512
nilpotency class	4
isoclinism	isoclinic class of #142975
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	D₄
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 459 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50173);