#139471

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 66 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)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18)(19,29)(20,34,37,54)(21,38,42,26)(22,27)(23,28)(24,31,48,39)(25,36,50,44)(30,59,62,79)(32,57)(33,58)(35,69)(40,68)(41,71,65,52)(43,74)(45,55)(46,75)(47,64,53,72)(49,70)(51,73)(56,61,76,66)(60,77)(63,67)(78,80);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,21,39)(18,24,38,58)(20,30,40,60)(22,43,41,64)(23,33,42,31)(25,46,49,76)(26,29,28,48)(27,53,71,57)(32,65,72,45)(34,67,68,59)(35,63,37,62)(36,66,73,75)(44,61,70,80)(47,52,74,55)(50,78,51,56)(54,77,69,79);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,20,23,37)(18,25,28,50)(19,30,33,62)(21,40,42,35)(22,44,45,36)(24,46,48,78)(26,51,38,49)(27,54,55,34)(29,56,58,76)(31,63,39,60)(32,66,43,61)(41,70,65,73)(47,67,53,77)(52,68,71,69)(57,79,74,59)(64,80,72,75)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=ghi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^c=ei,f^c=fghi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46667);