#72227029

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#72223651
commutator	C₄×C₃₂	C₂×C₄×C₈
Frattini, cosocle	C₁₆×M₆₄	C₂³
socle	C₂	#72227028
prime radical	#33981	M₃₂
LCS	G ▷ C₄×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₃₂ ◁ C₄×M₆₄ ◁ C₈×#1398 ◁ #72216231 ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^13|b^2=d,c^2=f,d^2=g,e^2=h,g^2=i,h^2=j,i^2=k,j^2=l,l^2=m,b^a=be,c^a=clm,c^b=cik,d^a=dh,d^c=dk,e^a=ehjlm,e^c=el,f^a=fm,f^b=fk,f^e=fm,g^a=gj,h^a=hjlm,h^c=hm,i^a=il,j^a=jlm,k^a=km,l^a=lm

Conjugacy classes & character table

404 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227170
maximal normal subgroup of	#72227170

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.12*F.13);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10*F.12*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.13);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.12*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);