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#72340468

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

3598 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 3598 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:

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

Polycyclic presentation:

2^11|a^2=i,b^2=ghij,c^2=j,d^2=hj,e^2=hj,f^2=k,b^a=bf,c^a=ce,c^b=ck,d^a=dek,d^b=dijk,d^c=dhk,e^a=eh,e^b=eg,e^c=ehjk,e^d=ehj,f^b=fik,f^c=fgi,f^d=fgi,f^e=fijk,g^a=gi,g^b=gjk,g^c=gj,g^d=gjk,g^e=gj,h^a=hk,h^b=hj,h^f=hk,i^b=ik,i^c=ij,i^d=ij,i^e=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340478
maximal normal subgroup of#72340478

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