Finite Group Databasebeta

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#72286024

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

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

normal subgroups only

This group has 646 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)(3,8,16,32)(4,17)(5,10,24,38)(6,11,28,40)(7,12)(9,19)(13,41,43,59)(14,30)(15,45)(18,50,46,48)(20,49)(21,36)(22,55)(23,37)(25,31)(26,57)(27,39)(29,56,58,42)(33,51,62,64)(34,54)(35,53)(44,61,47,60)(52,63);(1,3,7,16)(2,8)(4,18,21,46)(5,22)(6,25,28,15)(9,33)(10,30,38,55)(11,31)(12,32)(13,26,43,20)(14,24)(17,48,36,50)(19,51)(23,47)(27,44)(29,34)(35,62)(37,61,39,60)(40,45)(41,57,59,49)(42,54)(52,56)(53,64)(58,63);(1,4)(2,9)(3,13)(5,19)(6,26)(7,21)(8,29)(10,36)(11,34)(12,35)(14,42)(15,46)(16,43)(17,38)(18,25)(20,28)(22,56)(23,52)(24,53)(27,54)(30,59)(31,33)(32,58)(37,49)(39,57)(40,63)(41,55)(44,64)(45,62)(47,51)(48,60)(50,61)

Polycyclic presentation:

2^10|a^2=fg,b^2=gj,d^2=gh,e^2=gj,b^a=bfh,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dgh,e^a=ef,e^b=egj,e^c=egj,e^d=ehj,f^a=fij,f^b=fj,f^c=fh,f^d=fi,f^e=fij,g^a=gij,g^d=gj,h^a=hi,h^b=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286031, #72340229, #72340470, …
maximal normal subgroup of#72286031, #72340229, #72340470, …

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