#72227115

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227114
commutator	C₁₆×C₃₂	C₂²×C₄
Frattini, cosocle	C₁₆×M₆₄	C₂³
prime radical	#72225593	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

2994 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)(3,8,20,34)(4,9,26,38)(5,12,28,46)(6,11,27,16)(10,40,45,17)(13,23,48,19)(14,49,47,24)(18,43,53,42)(21,30,54,62)(22,32,50,29)(25,36,56,39)(31,58,35,52)(33,63,60,51)(37,59,41,57)(44,64,61,55);(3,17)(4,19)(5,24)(6,27)(8,10)(9,13)(11,16)(12,14)(18,50)(20,40)(21,55)(22,53)(23,26)(25,59)(28,49)(29,43)(30,44)(31,33)(32,42)(34,45)(35,60)(36,41)(37,39)(38,48)(46,47)(51,52)(54,64)(56,57)(58,63)(61,62);(2,8,11,29,15,34,16,32)(3,18,20,53)(4,23)(5,24)(6,27)(9,35,36,44,38,31,39,61)(10,41,42,13,45,37,43,48)(12,33,14,30,46,60,47,62)(17,50,40,22)(19,26)(21,58,54,52)(25,57)(28,49)(51,55,63,64)(56,59)

Polycyclic presentation:

2^13|a^2=m,c^2=f,d^2=gi,e^2=hj,f^2=k,g^2=ik,h^2=jl,i^2=k,j^2=lm,l^2=m,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dgk,e^a=eh,e^b=eh,e^c=ehl,f^a=fl,f^b=fk,f^e=fm,g^a=gj,g^b=gi,g^c=gi,h^a=hj,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

212 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, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 12, F.13);;
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);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.12);;
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, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
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);