Finite Group Databasebeta

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#72309199

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C8

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#13584
commutatorC4×C8C2×C42
Frattini, cosocleC2×#492C23
prime radicalC2×#3151C22
LCSG ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C22×C42 ◁ C2×#3151 ◁ G
LECSG ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1143 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1143 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) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,4,3)(6,11,12,13,15,19,14,20);(1,3,4,2)(5,6,8,19,10,15,7,11)(9,12,18,13,17,14,16,20);(7,16)(8,18)(9,17)(11,20)(12,14)(13,19)

Polycyclic presentation:

2^10|a^2=d,b^2=dej,d^2=ij,e^2=j,f^2=h,g^2=ij,h^2=j,b^a=bf,c^a=cgij,c^b=cgh,d^b=dhj,d^c=dij,e^a=eg,e^b=ehj,e^c=ehi,e^d=eij,f^b=fghj,f^c=fhj,f^e=fh,g^b=gh,g^c=gij,g^e=gij,h^b=hi,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72309200, #72309253, #72309260
maximal normal subgroup of#72309200, #72309253, #72309260

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