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#72284869

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#142537
commutator, Frattini, cosocle#2308C23
prime radical#23444C22
LCSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G
LECSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1102 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,18,22,49)(5,10)(6,11)(7,12)(8,17)(9,20,33,53)(14,41,43,58)(15,34)(16,40)(19,47,51,62)(21,48,54,64)(23,35)(24,36)(25,30)(26,37)(27,38)(28,56,39,42)(29,45)(31,61,63,50)(32,52,57,55)(44,60)(46,59);(1,3)(2,8,12,13)(4,19)(5,15)(6,25)(7,17)(9,31,33,63)(10,34,36,29)(11,30,38,40)(14,21)(16,27)(18,47,49,62)(20,50)(22,51)(23,46)(24,45)(26,44)(28,57,39,32)(35,60,37,59)(41,64,58,48)(42,52)(43,54)(53,61)(55,56);(1,4,6,21,7,22,27,54)(2,9,11,32)(3,14,16,19,17,43,25,51)(5,20,23,52,24,53,26,55)(8,28,30,31)(10,18,35,48)(12,33,38,57)(13,39,40,63)(15,42,44,50,45,56,46,61)(29,58,59,62)(34,41,60,47)(36,49,37,64)

Polycyclic presentation:

2^10|a^2=i,b^2=g,c^2=e,d^2=j,e^2=gj,b^a=bf,c^a=cd,c^b=ce,d^b=dg,d^c=dgh,e^a=eghj,e^b=egj,e^d=eij,f^b=fj,f^c=fhj,f^d=fj,f^e=fij,g^a=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284886
maximal normal subgroup of#72284886

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