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#72283564

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216457
commutator, Frattini, cosocle#1781C24
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3506 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 3506 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,8)(4,18)(5,11)(6,12)(7,13)(9,22)(10,17)(14,44)(15,32)(16,40)(19,52)(20,49)(21,50)(23,51)(24,56)(25,39)(26,42)(27,35)(28,33)(29,37)(30,43)(31,45)(34,60)(36,57)(38,54)(41,59)(46,48)(47,61)(53,64)(55,63)(58,62);(1,3)(2,8)(4,19)(5,15,27,25)(6,28)(7,17)(9,34,37,64)(10,13)(11,32,35,39)(12,33)(14,21)(16,30)(18,48)(20,24)(22,53,29,60)(23,46)(26,58,59,47)(31,36,38,63)(40,43)(41,61,42,62)(44,56)(45,55,54,57)(49,50)(51,52);(1,4,5,24,7,23,27,21)(2,9,35,63)(3,14,25,19)(6,29,59,55)(8,31,32,34,10,38,39,64)(11,36,13,37)(12,18,42,56,43,51,41,50)(15,46,17,20)(16,45,47,53,28,54,58,60)(22,26,57,30)(33,49,61,48)(40,44,62,52);(1,3)(2,10)(4,20)(5,25)(6,28)(7,17)(8,13)(9,31)(11,39)(12,40)(14,23)(15,27)(16,30)(18,49)(19,24)(21,46)(22,54)(26,58)(29,45)(32,35)(33,43)(34,63)(36,64)(37,38)(41,61)(42,62)(44,51)(47,59)(48,50)(52,56)(53,55)(57,60)

Polycyclic presentation:

2^11|b^2=gj,c^2=eghik,e^2=hik,f^2=h,b^a=bjk,c^a=cf,c^b=ce,d^a=dg,d^b=dgj,d^c=dgij,e^a=eg,e^b=eh,e^c=eik,e^d=eij,f^a=fh,f^b=fgi,f^c=fghk,f^d=fij,f^e=fik,g^c=gi,g^e=gj,g^f=gjk,h^b=hj,h^c=hjk,h^d=hk,h^e=hk,i^a=ij,i^b=ijk,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285103
maximal normal subgroup of#72285103

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