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#72341242

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485385
commutator, Frattini, cosocle#379C24
prime radical#23444C22
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G
LECSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1378 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1378 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)(3,13)(4,18,22,9)(5,10,25,35)(6,11,26,37)(7,12)(8,17)(14,27)(15,28,23,39)(16,29,46,36)(19,30,45,47)(20,33)(21,48)(24,53)(31,51)(32,40)(34,55)(38,42)(41,56,43,63)(44,57)(49,59)(50,60,54,61)(52,62)(58,64);(1,3,7,17)(2,8,12,13)(4,19,22,45)(5,23,25,15)(6,16,26,46)(9,30,18,47)(10,28,35,39)(11,36,37,29)(14,40,42,21)(20,49,51,64)(24,52,55,44)(27,48,38,32)(31,59,33,58)(34,62,53,57)(41,54,43,50)(56,60,63,61);(1,4)(2,9)(3,14,17,42)(5,20)(6,21,26,40)(7,22)(8,27)(10,33)(11,32,37,48)(12,18)(13,38)(15,43)(16,45,46,19)(23,41)(24,54,55,50)(25,51)(28,56,39,63)(29,47)(30,36)(31,35)(34,60,53,61)(44,64)(49,52)(57,58,62,59);(1,5)(2,10,12,35)(3,15)(4,20,22,51)(6,24)(7,25)(8,28,13,39)(9,31)(11,34,37,53)(14,41,42,43)(16,44)(17,23)(18,33)(19,49,45,64)(21,50,40,54)(26,55)(27,56)(29,57,36,62)(30,58)(32,60)(38,63)(46,52)(47,59)(48,61)

Polycyclic presentation:

2^10|a^2=ghi,b^2=j,c^2=fh,d^2=g,e^2=g,b^a=bj,c^a=cf,c^b=ce,d^a=dh,d^b=dgj,d^c=dhij,e^a=eh,e^b=eg,e^c=eghi,f^a=fij,f^b=fhi,f^c=fi,f^d=fij,f^e=fi,g^a=gj,g^c=gj,h^a=hj,h^b=hj,h^c=hi,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341245
maximal normal subgroup of#72341245

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