Finite Group Databasebeta

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#72284370

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

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

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

normal subgroups only

This group has 641 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)(3,8)(4,17)(5,10,23,36)(6,24)(7,12)(9,20,34,56)(11,26)(13,38)(14,28,44,62)(15,45)(16,30)(18,51)(19,32,55,35)(21,57,25,49)(22,50)(27,41,60,58)(29,47)(31,42,63,46)(33,39)(37,54)(40,59,53,61)(43,64)(48,52);(1,3,7,16)(2,8,12,30)(4,18,22,52)(5,14,23,44)(6,15,26,47)(9,31,34,63)(10,28,36,62)(11,29,24,45)(13,39,43,54)(17,48,50,51)(19,53,55,40)(20,46,56,42)(21,58,25,41)(27,57,60,49)(32,61,35,59)(33,38,37,64);(1,4,19,54,7,22,55,39)(2,9,32,49,12,34,35,57)(3,13,40,18,16,43,53,52)(5,20,26,21)(6,25,23,56)(8,27,59,31,30,60,61,63)(10,17,11,37)(14,41,47,42)(15,46,44,58)(24,33,36,50)(28,38,29,51)(45,48,62,64)

Polycyclic presentation:

2^10|a^2=fg,b^2=j,c^2=dehij,d^2=j,e^2=ghij,b^a=bi,c^a=cd,c^b=cdeghij,d^a=dhj,d^b=df,d^c=dfh,e^a=ef,e^b=efg,e^c=efh,e^d=ehij,f^a=fi,f^c=fh,f^e=fij,g^a=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj,i^c=ij

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284403, #72285311, #72285860, …
maximal normal subgroup of#72284403, #72285311, #72285860, …

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