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#72284400

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 ◁ #20585 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3643 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 3643 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)(6,28)(7,13)(9,22)(10,17)(12,30)(14,44)(15,32)(16,33)(19,52)(20,50)(21,51)(23,42)(24,57)(25,39)(26,41)(27,35)(29,40)(31,45)(34,62)(36,58)(37,56)(38,48)(43,55)(46,49)(47,61)(53,64)(54,63)(59,60);(1,3)(2,8)(4,19)(5,15,27,25)(6,29)(7,17)(9,34,37,64)(10,13)(11,32,35,39)(12,33)(14,21)(16,30)(18,49)(20,24)(22,53,56,62)(23,46)(26,59,55,47)(28,40)(31,36,38,63)(41,60,43,61)(42,52)(44,57)(45,54,48,58)(50,51);(1,4,5,24,7,23,27,21)(2,9,35,63)(3,14,25,19)(6,22,55,58)(8,31,32,34,10,38,39,64)(11,36,13,37)(12,42,43,51,28,18,41,57)(15,46,17,20)(16,48,47,62,29,45,59,53)(26,54,30,56)(33,44,60,52)(40,50,61,49);(1,3)(2,10)(4,20)(5,25)(6,29)(7,17)(8,13)(9,31)(11,39)(12,40)(14,23)(15,27)(16,30)(18,50)(19,24)(21,46)(22,48)(26,59)(28,33)(32,35)(34,63)(36,64)(37,38)(41,60)(42,44)(43,61)(45,56)(47,55)(49,51)(52,57)(53,54)(58,62)

Polycyclic presentation:

2^11|b^2=gj,c^2=eghik,e^2=hik,f^2=hk,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=fhk,f^b=fgi,f^c=fgh,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.

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*F.11);;
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*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
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);