#72319632

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	#72317668
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318239	C₂
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

4292 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:

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

Polycyclic presentation:

2^11|b^2=ghk,c^2=f,d^2=hk,f^2=j,i^2=k,b^a=bg,c^a=cf,c^b=ce,d^a=dhk,d^b=dh,d^c=defgijk,e^a=eh,e^b=ehik,e^c=eij,e^d=eh,f^a=fj,f^b=fijk,f^d=fh,f^e=fjk,g^c=ghik,g^e=gjk,g^f=gj,h^c=hk,i^a=ij,i^b=ij,i^c=ij,i^e=ik,i^g=ik,j^b=jk

Conjugacy classes & character table

65 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.11);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8*F.11);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, 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);;
gap> SetConjugate(coll, 9, 5, F.9*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);