Finite Group Databasebeta

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#72283880

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

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

normal subgroups only

This group has 667 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)(3,8,16,31)(4,17)(5,10,24,37)(6,25,27,11)(7,12)(9,19)(13,39)(14,35)(15,30)(18,51,45,48)(20,49,26,59)(21,50)(22,34)(23,36)(28,42)(29,44)(32,52,62,55)(33,54,38,53)(40,60)(41,64)(43,61,46,63)(47,57)(56,58);(1,3,7,16)(2,8,12,31)(4,18)(5,14,24,44)(6,15)(9,32)(10,35,37,29)(11,30)(13,26,41,20)(17,48,50,51)(19,52,22,55)(21,45)(23,46)(25,57)(27,47)(28,38,60,33)(34,62)(36,61)(39,49)(40,54)(42,53)(43,56)(58,63)(59,64);(1,4,7,21)(2,9,12,34)(3,13)(5,22)(6,26,27,20)(8,28)(10,17)(11,38,25,33)(14,42,44,40)(15,45)(16,41)(18,47)(19,24)(23,53)(29,39,35,64)(30,62)(31,60)(32,57)(36,59)(37,50)(43,52,46,55)(48,63,51,61)(49,58)(54,56)

Polycyclic presentation:

2^10|a^2=fg,b^2=fghij,c^2=hj,d^2=gh,b^a=bghj,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dghi,e^a=efhi,e^b=ehj,e^c=ej,e^d=ehij,f^a=fij,f^b=fi,f^c=fh,f^d=fi,g^a=gij,g^b=gij,g^d=gj,g^e=gj,h^a=hi,h^b=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283923, #72284062
maximal normal subgroup of#72283923, #72284062

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