Finite Group Databasebeta

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#72338815

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#251827
commutator, Frattini, cosocleC22×#127C24
LCSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1

5025 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 5025 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|c^2=j,f^2=k,g^2=j,c^a=cg,c^b=ce,d^a=defk,d^b=dek,d^c=dek,e^a=eh,f^a=fh,f^b=fhk,f^c=fi,f^e=fk,g^a=gj,g^b=ghj,g^c=gj,g^d=ghik,g^e=gj,h^c=hj,h^d=hk,i^a=ij,i^b=ijk,i^d=ik

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338830, #72339103, #72340564
maximal normal subgroup of#72338830, #72339103, #72340564

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);