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#72285485

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485605
commutator, Frattini, cosocle#381C24
prime radical#7266800C2
LCSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G
LECSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1526 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=gh,c^2=ehj,d^2=hi,e^2=h,f^2=j,b^a=bghj,c^a=ce,c^b=cf,d^a=dg,d^b=dg,d^c=dgh,e^a=eh,e^b=eg,e^d=ei,f^a=fgj,f^b=fij,f^c=fg,f^d=fi,f^e=fi,g^b=gj,g^c=gi,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285567
maximal normal subgroup of#72285567

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