#72227260

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227259
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	C₂×#508	#106
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

2^12|a^2=d,c^2=dfghj,d^2=g,e^2=hikl,f^2=i,g^2=kl,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=ce,c^b=cfl,d^b=di,d^c=dij,e^a=eh,e^b=eik,e^c=ehikl,f^a=fkl,f^b=fikl,f^c=fil,f^d=fl,g^b=gk,g^c=gkl,h^a=hj,h^b=hl,h^c=hjl,i^a=il,i^b=ikl,i^c=ik,j^a=jl,j^c=jl,k^b=kl,k^c=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227291
maximal normal subgroup of	#72227291

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