#131703

order	512
nilpotency class	4
isoclinism	isoclinic class of #131671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62489	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=egi,f^c=fgi

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, 38899);