#131738

order	512
nilpotency class	4
isoclinism	isoclinic class of #131668
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	#62341	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

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

normal subgroups only

This group has 551 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)(17,18,23,29,22,28,46,27,24,30,49,61,45,60,21,43)(19,31,35,53,34,58,73,62,36,63,75,25,47,52,33,51)(20,37,40,59,42,26,44,55,41,57,48,54,39,56,38,64)(32,66,69,78,71,65,72,50,70,79,74,76,68,77,67,80);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,24,36)(18,25,30,53)(20,32,41,70)(21,35,46,75)(22,47,45,34)(23,33,49,73)(26,50,56,80)(27,58,43,52)(28,62,60,51)(29,63,61,31)(37,76,57,78)(38,69,44,74)(39,71,42,68)(40,67,48,72)(54,66,59,79)(55,65,64,77);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20)(18,26)(19,32)(21,44)(22,42)(23,48)(24,41)(25,50)(27,59)(28,57)(29,64)(30,56)(31,65)(33,72)(34,71)(35,74)(36,70)(37,60)(38,46)(39,45)(40,49)(43,54)(47,68)(51,76)(52,66)(53,80)(55,61)(58,79)(62,78)(63,77)(67,73)(69,75)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=g,f^2=ghi,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=eg,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, 38934);