Finite Group Databasebeta

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#72283743

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#150393
commutator#379C22×C4
Frattini, cosocle#1781C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1057 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 1057 conjugacy classes of subgroups — too many to draw. Only its 55 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)(3,13)(4,18)(5,10,24,34)(6,11,25,35)(7,12)(8,15)(9,21)(14,36,41,55)(16,44)(17,38)(19,30,22,32)(20,31,50,47)(23,33)(26,39,29,51)(27,42)(28,43)(37,46)(40,61,63,60)(45,54)(48,58)(49,57)(52,59)(53,62,56,64);(1,3,6,16)(2,8,11,27,12,28,35,54)(4,14,20,40,21,41,50,63)(5,15,23,42,24,43,52,45)(7,17,25,46)(9,29,31,56)(10,13,33,37)(18,26,47,53)(19,39,48,62)(22,51,49,64)(30,55,57,61,32,36,58,60)(34,38,59,44);(1,4,7,21)(2,9)(3,14,17,41)(5,22,24,19)(6,20)(8,26)(10,32)(11,31,35,47)(12,18)(13,36)(15,39,43,51)(16,40)(23,48)(25,50)(27,53,54,56)(28,29)(30,34)(33,57,59,58)(37,61,44,60)(38,55)(42,64)(45,62)(46,63)(49,52)

Polycyclic presentation:

2^10|a^2=fh,b^2=e,c^2=ghj,d^2=ghi,e^2=gi,b^a=bd,c^a=chi,c^b=cfghi,d^a=dg,d^b=df,d^c=dij,e^a=efgh,e^c=ehi,e^d=eh,f^a=fi,f^b=fh,f^c=fij,f^d=fi,f^e=fi,g^a=gij,g^b=gj,g^c=gj,g^d=gj,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283811, #72284129, #72285637, …
maximal normal subgroup of#72283811, #72284129, #72285637, …

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