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#72306608

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13673
commutator, Frattini, cosocle#493C24
prime radical#488869C2
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

1379 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1379 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)(5,22)(6,12)(7,13)(8,14)(9,19)(11,27)(16,35)(17,44)(18,37)(20,40)(21,39)(23,38)(24,55)(25,56)(26,49)(28,36)(29,60)(30,61)(31,47)(32,46)(33,48)(34,43)(41,58)(42,59)(45,57)(50,52)(51,62)(53,63)(54,64);(1,3,8,4,7,19,33,18)(2,9,14,37,13,15,48,10)(5,23)(6,28,32,53,31,17,30,21)(11,40)(12,44,46,39,47,36,61,63)(16,26)(20,27)(22,38)(24,29)(25,54)(34,52)(35,49)(41,45)(42,62)(43,50)(51,59)(55,60)(56,64)(57,58);(1,4,8,19,7,18,33,3)(2,10,14,9,13,37,48,15)(5,24)(6,21,32,28,31,53,30,17)(11,41)(12,39,46,36,47,63,61,44)(16,25)(20,29)(22,55)(23,57)(26,51)(27,58)(34,54)(35,56)(38,45)(40,60)(42,50)(43,64)(49,62)(52,59);(1,5)(2,11)(3,16)(4,20)(6,29)(7,27)(8,34)(9,35)(10,38)(12,45)(13,22)(14,49)(15,50)(17,24)(18,23)(19,52)(21,54)(25,30)(26,33)(28,58)(31,57)(32,59)(36,55)(37,40)(39,62)(41,44)(42,61)(43,48)(46,56)(47,60)(51,53)(63,64)

Polycyclic presentation:

2^10|b^2=ghj,c^2=ghj,e^2=h,g^2=i,h^2=j,b^a=bij,c^b=cj,d^a=dj,d^b=df,d^c=de,e^b=eg,e^d=ehj,f^a=fj,f^b=fh,f^c=fgh,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

58 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.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.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, 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);;
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);