Finite Group Databasebeta

← Prev (#72284830) Next (#72284832) →

#72284831

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139504
commutatorC22×C22⋊C4C22×C4
Frattini, cosocle#2308C23
prime radical#417825C2
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35446 ◁ G
LECSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1841 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 1841 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,13,17,8)(4,18,22,51)(5,23)(6,11,28,40)(7,12)(9,20)(10,25)(14,42,44,59)(15,30,46,35)(16,31)(19,52,47,49)(21,50,58,63)(24,36,27,39)(26,37)(29,43)(32,54)(33,57)(34,56)(38,55)(41,64)(45,60)(48,61)(53,62);(1,3)(2,8,12,13)(4,19,22,47)(5,15,25,46)(6,26)(7,17)(9,32,34,62)(10,35)(11,37,40,31)(14,21)(16,28)(18,49,51,52)(20,54,56,53)(23,30)(24,45,27,48)(29,38)(33,41)(36,61)(39,60)(42,50)(43,57)(44,58)(55,64)(59,63);(1,4,6,21)(2,9,11,38,12,34,40,33)(3,14,26,19,17,44,16,47)(5,20,27,55)(7,22,28,58)(8,29,37,62)(10,18,39,63,23,51,36,50)(13,41,31,32)(15,43,45,54,46,64,48,53)(24,57,25,56)(30,59,60,49)(35,42,61,52)

Polycyclic presentation:

2^10|a^2=f,b^2=fh,c^2=efgi,d^2=ghj,e^2=gh,b^a=bhj,c^a=cd,c^b=ce,d^a=dg,d^b=dh,d^c=dg,e^a=ehi,e^b=egi,e^c=ehi,e^d=ej,f^b=fij,f^c=fhj,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gi,g^d=gj,g^e=gj,h^a=hij,h^b=hij,h^c=hij,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284886, #72285041
maximal normal subgroup of#72284886, #72285041

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