Finite Group Databasebeta

← Prev (#72340936) Next (#72340938) →

#72340937

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151462
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

427 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 37 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)(3,13,17,34)(4,18,22,38)(5,10)(6,25,27,11)(7,12)(8,15,30,46)(9,21)(14,41)(16,40)(19,50)(20,52,54,49)(23,36,56,58)(24,37)(26,33)(28,59,31,62)(29,45)(32,55)(35,53)(39,51,63,43)(42,48)(44,60)(47,61)(57,64);(1,3)(2,8)(4,19)(5,15,24,46)(6,16,27,47)(7,17)(9,31,33,28)(10,34)(11,29)(12,30)(13,37)(14,22)(18,48,38,60)(20,44,54,42)(21,51,26,43)(23,45)(25,57)(32,39,35,63)(36,40)(41,52,50,49)(53,59)(55,62)(56,64)(58,61);(1,4,7,22)(2,9,12,33)(3,14)(5,20)(6,26,27,21)(8,28,30,31)(10,35)(11,38,25,18)(13,39)(15,44,46,42)(16,43)(17,19)(23,53)(24,54)(29,60,57,48)(32,37)(34,63)(36,52)(40,41)(45,62,64,59)(47,51)(49,58)(50,61)(55,56)

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=fhj,d^2=fj,e^2=fhj,b^a=bd,c^a=ce,c^b=ch,d^a=df,d^b=dfh,d^c=dfg,e^a=efhi,e^b=efgh,e^c=efhj,e^d=eh,f^b=fj,f^c=fij,f^e=fj,g^a=gh,g^c=gi,g^d=gij,g^e=gij,h^b=hi,h^c=hij,h^d=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340939, #72341200, #72341228
maximal normal subgroup of#72340939, #72341200, #72341228

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