Finite Group Databasebeta

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#72339611

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144386
commutator, Frattini, cosocleC2×#535C23
prime radical#417790C2
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1957 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1957 conjugacy classes of subgroups — too many to draw. Only its 61 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=dhij,d^2=i,f^2=j,g^2=j,h^2=j,b^a=bf,c^a=cd,c^b=ce,d^b=dh,d^c=dij,e^a=eg,e^c=ehj,e^d=eij,f^b=fj,f^c=fghi,f^d=fj,f^e=fj,g^b=gj,g^c=gi,g^e=gj,h^a=hj,h^b=hj,h^c=hij,h^e=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339616
maximal normal subgroup of#72339616
maximal quotient of#72339620

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