#72319691

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72317683
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#246927	C₂²
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35327 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=ghij,c^2=f,d^2=hijk,e^2=j,f^2=jk,h^2=k,i^2=k,b^a=bg,c^a=ce,c^b=ceghi,d^b=dhij,d^c=defhjk,e^a=ej,e^b=eh,e^c=ehik,e^d=ehij,f^a=fhijk,f^b=fi,f^d=fhij,f^e=fk,g^c=gh,g^e=gj,g^f=gjk,h^a=hj,h^b=hjk,h^c=hj,h^g=hk,i^a=ij,i^b=ij,i^c=ijk,i^g=ik,j^b=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320243
maximal normal subgroup of	#72320243

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