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#72283594

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

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

normal subgroups only

This group has 672 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)(3,8)(4,17)(5,23,19,10)(6,11)(7,12)(9,21)(13,43)(14,37,24,31)(15,32)(16,33)(18,50,52,48)(20,53,25,49)(22,40)(26,39,28,41)(27,59)(29,42)(30,44)(34,51,62,55)(35,54,38,58)(36,56)(45,64)(46,61,57,63)(47,60);(1,3)(2,8,12,33)(4,18)(5,24)(6,27,29,15)(7,16)(9,34,36,62)(10,37,23,31)(11,32)(13,20,45,25)(14,19)(17,48,40,50)(21,55)(22,52)(26,57,28,46)(30,38)(35,60)(39,61)(41,63)(42,59)(43,53)(44,58,47,54)(49,64)(51,56);(1,4,19,20,7,22,5,25)(2,9,10,38,12,36,23,35)(3,13,14,18)(6,21,26,58)(8,30,37,62)(11,40,41,53)(15,47,46,55,27,44,57,51)(16,45,24,52)(17,39,49,42)(28,54,29,56)(31,34,33,60)(32,43,63,50,59,64,61,48)

Polycyclic presentation:

2^10|a^2=g,b^2=fi,c^2=dfghj,d^2=ghj,e^2=gj,b^a=bghij,c^a=ce,c^b=cd,d^a=df,d^b=dg,d^c=dhj,e^a=egi,e^b=efh,e^c=efgj,e^d=ehj,f^a=fj,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

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283692, #72340887, #72341219, …
maximal normal subgroup of#72283692, #72340887, #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);;
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*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*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.9);;
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, 1, F.6*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);