#131731

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
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	#62325	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38927);