Finite Group Databasebeta

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#72303976

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8535
commutatorC23×D4C22×C4
Frattini, prime radical, cosocleC22×#121C23
LCSG ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G
LECSG ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1

1374 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 32
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,9,11,7)(4,8)(5,6)(10,16,13,15)(12,14)(17,18)(19,22)(20,25,28,23)(21,24)(26,29)(27,31,32,30);(2,6)(3,10)(4,12)(7,15)(9,16)(11,13)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,32,30);(1,3)(2,7)(4,13)(5,11)(6,9)(8,15)(10,12)(14,16)(17,20,21,27,22,28,29,32)(18,23,24,30,19,25,26,31)

Polycyclic presentation:

2^10|a^2=gh,b^2=j,c^2=d,d^2=j,e^2=i,b^a=bghij,c^a=cf,c^b=ce,d^b=dg,e^a=eh,e^b=ei,e^c=egi,e^d=ej,f^a=fij,f^b=fhj,f^e=fi,g^c=gj,h^c=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303984
maximal normal subgroup of#72303984

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