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#72306609

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13674
commutator, Frattini, cosocle#493C24
prime radical#61105C22
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1337 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1337 conjugacy classes of subgroups — too many to draw. Only its 115 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)(4,10)(5,23)(6,12)(7,14)(8,13)(9,20)(11,29)(16,35)(17,36)(18,51)(19,38)(21,41)(22,40)(24,39)(25,58)(26,59)(27,45)(28,44)(30,46)(31,61)(32,62)(33,48)(34,49)(37,55)(42,60)(43,52)(47,56)(50,54)(53,63)(57,64);(1,3,16,19,8,20,7,4)(2,9,35,10,13,15,14,38)(5,24,29,21)(6,18,32,30,34,55,33,22)(11,41,23,39)(12,37,62,40,49,51,48,46)(17,27,54,28)(25,31,60,56)(26,53,52,57)(36,45,50,44)(42,47,58,61)(43,64,59,63);(1,4,16,3,8,19,7,20)(2,10,35,15,13,38,14,9)(5,25)(6,30,32,55,34,22,33,18)(11,42)(12,46,62,37,49,40,48,51)(17,52)(21,56)(23,58)(24,31)(26,54)(27,57)(28,53)(29,60)(36,43)(39,61)(41,47)(44,63)(45,64)(50,59);(1,5)(2,11)(3,17)(4,21)(6,31)(7,27)(8,29)(9,36)(10,39)(12,47)(13,23)(14,44)(15,50)(16,28)(18,25)(19,24)(20,54)(22,57)(26,32)(30,53)(33,52)(34,56)(35,45)(37,58)(38,41)(40,63)(42,51)(43,62)(46,64)(48,59)(49,61)(55,60)

Polycyclic presentation:

2^10|b^2=ghj,c^2=ghij,e^2=h,g^2=i,h^2=j,b^a=bij,c^b=cij,d^a=dj,d^b=df,d^c=de,e^b=eg,e^c=ej,e^d=ehj,f^a=fj,f^b=fh,f^c=fghj,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306617
maximal normal subgroup of#72306617

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