Finite Group Databasebeta

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#72283264

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152185
commutator#492C22×C4
Frattini, cosocle#3147C23
prime radical#6367523C2
LCSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1015 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 75 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:

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

Polycyclic presentation:

2^10|a^2=gi,b^2=gj,c^2=dfj,e^2=h,f^2=j,g^2=ij,h^2=j,b^a=bij,c^a=cd,c^b=ce,d^a=dh,d^b=dg,e^a=eghj,e^b=ej,e^c=eh,e^d=eh,f^a=fh,f^b=fgj,f^e=fhj,g^c=gh,g^d=gij,g^f=gij,h^a=hi,h^c=hj,h^d=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283316, #72284059
maximal normal subgroup of#72283316, #72284059

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