Finite Group Databasebeta

← Prev (#72283570) Next (#72283572) →

#72283571

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151391
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

453 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 39 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)(4,17,21,37)(5,10,24,39)(6,25)(7,12)(9,19,35,56)(11,28)(13,42,44,64)(14,36,22,30)(15,40)(16,32)(18,51)(20,57)(23,41,27,38)(26,50)(29,43,60,45)(31,48)(33,52)(34,58)(46,61,47,63)(49,53)(54,62)(55,59);(1,3,7,16)(2,8,12,32)(4,18)(5,22,24,14)(6,15)(9,33)(10,36,39,30)(11,40)(13,26)(17,49,37,51)(19,54,56,52)(20,44)(21,53)(23,47)(25,31)(27,46)(28,48)(29,59)(34,60)(35,62)(38,61)(41,63)(42,57,64,50)(43,58,45,55);(1,4)(2,9,12,35)(3,13,16,44)(5,19)(6,26,28,20)(7,21)(8,29)(10,37,39,17)(11,34)(14,45,22,43)(15,18)(23,55,27,58)(24,56)(25,59)(30,42)(31,62,40,33)(32,60)(36,64)(38,50)(41,57)(46,54)(47,52)(48,53)(49,63,51,61)

Polycyclic presentation:

2^10|a^2=fi,b^2=fhij,c^2=gh,d^2=ghj,e^2=gj,b^a=bhi,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dgh,e^a=efh,e^b=egh,e^c=egi,e^d=ehj,f^b=fi,f^c=fh,f^d=fij,f^e=fi,g^a=gij,g^b=gj,g^d=gj,h^a=hi,h^b=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283653, #72285713
maximal normal subgroup of#72283653, #72285713

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