Finite Group Databasebeta

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#72321469

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#510507
commutator, Frattini, cosocleC22×C16C24
prime radical#500850C2
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ #33432 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

782 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 133 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=efh,d^2=efghj,e^2=hij,f^2=j,h^2=ij,i^2=j,b^a=bg,c^a=cfgi,c^b=ce,d^a=dfij,d^b=defij,d^c=di,e^b=eh,e^c=ej,e^d=ej,f^c=fj,g^c=gj,g^d=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321482
maximal normal subgroup of#72321482

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