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#72306755

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#512862
commutator, Frattini, cosocle#493C24
prime radical#489351C2
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1372 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 113 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:

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

Polycyclic presentation:

2^10|a^2=ghj,b^2=i,d^2=j,e^2=h,g^2=i,h^2=j,b^a=bi,c^a=ci,c^b=cij,d^a=de,d^b=df,d^c=dj,e^b=eg,e^c=ej,e^d=ehj,f^a=fghj,f^b=fj,f^c=fj,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#72306759
maximal normal subgroup of#72306759

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, 1, F.7*F.8*F.10);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, 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);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
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.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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);