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#72341193

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151462
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

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

normal subgroups only

This group has 423 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)(3,13,17,41)(4,18)(5,10)(6,25)(8,29)(9,33,36,20)(11,28)(14,42,46,64)(15,47)(16,32)(19,39)(21,52,26,58)(22,53)(23,37,27,57)(24,38)(30,59,51,54)(31,48,62,50)(34,60)(35,43)(40,49)(44,61)(45,63)(55,56);(1,3,6,16)(2,8,11,13,12,32,25,41)(4,19,26,59,22,44,21,54)(5,15,23,48,24,49,27,50)(7,17,28,29)(9,34,56,64,36,63,35,42)(10,31,37,47)(14,43,60,20)(18,51,58,61)(30,52,39,53)(33,46,55,45)(38,62,57,40);(1,4,7,22)(2,9,12,36)(3,14)(5,20,24,33)(6,26)(8,30,32,51)(10,18,38,53)(11,35)(13,39)(15,44)(16,45,29,60)(17,46)(19,49)(21,28)(23,43)(25,56)(27,55)(31,34,62,63)(37,52)(40,42)(41,61)(47,64)(48,59,50,54)(57,58)

Polycyclic presentation:

2^10|a^2=ghij,b^2=efij,c^2=hj,e^2=gij,b^a=be,c^a=cd,c^b=cdgij,d^a=dij,d^b=df,e^a=eg,e^b=ehij,e^c=efj,e^d=ehi,f^a=fh,f^b=fhi,f^c=fj,f^e=fij,g^a=gj,g^c=gij,g^d=gj,h^b=hi,h^e=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341200, #72341219
maximal normal subgroup of#72341200, #72341219

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