#139677

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,e^b=eh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46873);