Finite Group Databasebeta

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#72317966

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#147779
commutator#532C22×C4
Frattini, cosocleC2×#532C23
prime radical#7503371C2
LCSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G
LECSG ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

2447 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 2447 conjugacy classes of subgroups — too many to draw. Only its 147 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=efgi,e^2=j,f^2=i,h^2=j,b^a=be,c^a=cefi,c^b=cd,d^a=dh,d^c=dj,e^a=eh,e^b=ej,f^a=fh,f^b=fhj,f^d=fij,f^e=fij,g^b=gh,g^d=gij,g^e=gij,g^f=gij,h^a=hi,h^b=hj,h^c=hi,h^d=hj,h^e=hj,h^g=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72317985, #72318047, #72318165
maximal normal subgroup of#72317985, #72318047, #72318165

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