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#72340887

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237736
commutator, Frattini, cosocle#1781C24
prime radical#72258232C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C26 ◁ #21746 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2398 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 2398 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

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

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

Polycyclic presentation:

2^11|b^2=k,c^2=h,d^2=hik,e^2=h,f^2=hik,b^a=bhi,c^a=ck,c^b=ce,d^b=df,d^c=dgi,e^a=ejk,e^b=eh,e^c=eh,e^d=eg,f^b=fhik,f^c=fg,f^d=fhik,f^e=fi,g^b=gi,g^c=gk,g^d=gk,g^e=gjk,g^f=gj,h^d=hjk,h^f=hk,i^c=ij,i^d=ijk,i^e=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340897

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