Finite Group Databasebeta

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#72342915

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#486293
commutator, Frattini, cosocleC2×M32C24
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

1239 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 151 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:

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

Polycyclic presentation:

2^10|c^2=gj,e^2=j,f^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cgj,c^b=ce,d^c=dg,e^a=eg,e^b=ej,e^d=ej,f^a=fh,f^b=fh,f^c=fg,f^e=fj,g^b=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342917, #72342920, #72342980
maximal normal subgroup of#72342917, #72342920, #72342980

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