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#72285604

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151342
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

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

normal subgroups only

This group has 618 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=fi,c^2=dfghj,d^2=ghj,e^2=g,b^a=bij,c^a=ce,c^b=cd,d^a=df,d^b=dg,d^c=dhj,e^a=egj,e^b=efh,e^c=efgj,e^d=ehj,f^c=fh,f^d=fi,f^e=fij,g^b=gi,g^c=gij,g^d=gj,h^a=hi,h^b=hij,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285637
maximal normal subgroup of#72285637

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