#143129

order	512
nilpotency class	4
isoclinism	isoclinic class of #143090
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,27)(19,29,33,60)(20,34,37,25)(21,39,44,58)(22,45,42,28)(24,47,49,31)(26,53,56,43)(30,61,35,59)(32,65,46,54)(36,69,41,74)(38,52,71,67)(40,57,73,50)(48,64,66,62)(51,72,55,76)(63,80,68,78)(70,77,75,79);(6,15)(9,16)(17,19,23,33)(18,24,27,49)(20,35,37,30)(21,40,44,73)(22,46,42,32)(25,48,34,66)(26,54,56,65)(28,57,45,50)(29,58,60,39)(31,53,47,43)(36,70,41,75)(38,63,71,68)(51,78,55,80)(52,79,67,77)(59,74,61,69)(62,72,64,76);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,22,38,23,37,42,71)(18,25,28,52,27,34,45,67)(19,30,32,63,33,35,46,68)(21,41,43,76,44,36,53,72)(24,48,50,77,49,66,57,79)(26,55,39,69,56,51,58,74)(29,59,54,80,60,61,65,78)(31,64,40,70,47,62,73,75)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50325);