#72227127

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227126
commutator	C₁₆×C₃₂	C₂²×C₄
Frattini, cosocle	C₁₆×M₆₄	C₂³
prime radical	#72225451	C₂
LCS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^13|a^2=m,b^2=f,d^2=gik,e^2=hjl,g^2=ik,h^2=jlm,i^2=k,j^2=lm,l^2=m,b^a=bl,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dgk,e^a=ehl,e^b=ehm,e^c=ehl,f^c=fk,f^e=fm,g^a=gj,g^b=gi,g^c=gi,h^a=hjm,h^b=hj,h^c=hjm,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl,k^a=km,l^a=lm,l^b=lm,l^c=lm

Conjugacy classes & character table

200 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, 1, F.13);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
gap> SetPower(coll, 5, F.8*F.10*F.12);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12*F.13);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.12);;
gap> SetConjugate(coll, 6, 3, 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, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);