#143121

order	512
nilpotency class	4
isoclinism	isoclinic class of #143098
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

337 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 64 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)(19,29,32,58)(20,34,37,25)(21,39,43,56)(22,44,41,28)(24,46,48,31)(26,52,54,42)(30,57,35,65)(33,61,45,53)(36,68,71,73)(38,51,70,64)(40,55,75,49)(47,66,62,59)(50,72,63,79)(60,80,67,77)(69,76,74,78);(6,15)(9,16)(17,19,23,32)(18,24,27,48)(20,35,37,30)(21,40,43,75)(22,45,41,33)(25,47,34,62)(26,53,54,61)(28,55,44,49)(29,56,58,39)(31,52,46,42)(36,69,71,74)(38,60,70,67)(50,77,63,80)(51,78,64,76)(57,68,65,73)(59,72,66,79);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,22,38,23,37,41,70)(18,25,28,51,27,34,44,64)(19,30,33,60,32,35,45,67)(21,36,42,72,43,71,52,79)(24,47,49,76,48,62,55,78)(26,50,39,73,54,63,56,68)(29,57,53,77,58,65,61,80)(31,59,40,74,46,66,75,69)

Polycyclic presentation:

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

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, 50317);