#131728

order	512
nilpotency class	4
isoclinism	isoclinic class of #131660
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	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 544 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)(17,18,23,29,22,31,47,27,24,30,48,57,46,28,21,45)(19,32,36,52,38,56,68,60,37,59,67,25,35,51,34,50)(20,39,42,53,44,55,73,71,43,58,72,74,41,26,40,54)(33,62,65,76,64,61,79,75,66,77,80,70,69,78,63,49);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,24,37)(18,25,30,52)(20,33,43,66)(21,36,47,67)(22,35,46,38)(23,34,48,68)(26,49,55,75)(27,56,45,51)(28,50,31,60)(29,59,57,32)(39,70,58,76)(40,65,73,80)(41,64,44,69)(42,63,72,79)(53,77,74,62)(54,78,71,61);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20)(18,26)(19,33)(21,40)(22,44)(23,42)(24,43)(25,49)(27,53)(28,58)(29,54)(30,55)(31,39)(32,61)(34,63)(35,69)(36,65)(37,66)(38,64)(41,46)(45,74)(47,73)(48,72)(50,76)(51,62)(52,75)(56,77)(57,71)(59,78)(60,70)(67,80)(68,79)

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,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fgi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38924);