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#72284311

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151196
commutator, Frattini, cosocle#1781C23
prime radical#23895C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

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

normal subgroups only

This group has 699 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:

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

Polycyclic presentation:

2^10|a^2=fhj,b^2=fhij,c^2=dfgij,d^2=ghi,e^2=gi,b^a=bfh,c^a=ce,c^b=cd,d^a=df,d^b=dg,d^c=dhi,e^a=eghj,e^b=efi,e^c=efgij,e^d=eh,f^a=fi,f^b=fij,f^c=fh,f^d=fij,f^e=fi,g^a=gj,g^b=gi,g^c=gij,g^d=gj,h^a=hi,h^b=hij,h^c=hij,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284400, #72340298, #72340468, …
maximal normal subgroup of#72284400, #72340298, #72340468, …

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