Finite Group Databasebeta

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#72318117

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144412
commutator, Frattini, cosocleC2×#535C23
prime radicalC2×#3021C22
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1537 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1537 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)(3,8)(4,18,22,54)(5,24,26,10)(6,27,30,41)(7,12)(9,21,39,59)(11,31)(13,45,19,56)(14,33,16,35)(15,50)(17,36)(20,55,23,43)(25,44)(28,34,49,60)(29,38,58,40)(32,46,37,61)(42,52)(47,64,48,53)(51,62,57,63);(1,3,7,17)(2,8,12,36)(4,19)(5,14,26,16)(6,28,30,49)(9,37)(10,33,24,35)(11,42,44,50)(13,22)(15,31,52,25)(18,53)(20,48)(21,51)(23,47)(27,60,41,34)(29,46)(32,39)(38,63)(40,62)(43,45)(54,64)(55,56)(57,59)(58,61);(1,4,7,23)(2,9,12,40)(3,13,17,48)(5,22,26,20)(6,29,30,21)(8,32,36,63)(10,39,24,38)(11,43,44,18)(14,47,16,19)(15,51,52,46)(25,58,31,59)(27,55,41,54)(28,61,49,57)(33,62,35,37)(34,53,60,45)(42,64,50,56)

Polycyclic presentation:

2^10|a^2=f,b^2=hij,c^2=j,d^2=j,e^2=j,g^2=j,b^a=bfhi,c^a=ce,c^b=cd,d^a=dh,d^c=dj,e^a=eg,e^b=eghij,e^c=ej,f^c=fg,f^d=fj,f^e=fij,g^a=gi,g^b=gj,g^c=gj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318132, #72318183, #72318343, …
maximal normal subgroup of#72318132, #72318183, #72318343, …

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