Finite Group Databasebeta

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#72338788

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C4

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

5215 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 5215 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)(3,13)(4,15)(5,10)(6,11)(7,12)(8,14)(9,16)(19,27)(23,31)(28,32)(29,30);(3,9)(4,8)(17,18)(19,28)(20,24)(21,25)(22,26)(23,29)(27,31)(30,32);(1,3,7,9)(2,8,12,4)(5,15)(6,14)(10,16)(11,13)(17,19,22,29)(18,23,26,28)(20,30)(21,27)(24,32)(25,31);(1,4)(2,9)(3,12)(5,13)(6,16)(7,8)(10,15)(11,14)(19,29)(23,28)(24,25)(27,30)

Polycyclic presentation:

2^11|c^2=efj,f^2=j,g^2=jk,b^a=bh,c^a=ce,c^b=cefj,d^a=dg,d^b=defjk,d^c=dj,e^b=eh,e^c=eh,e^d=ehi,f^a=fh,f^b=fh,f^c=fh,f^d=fhi,f^e=fj,g^a=gjk,g^b=gh,g^c=ghijk,g^d=gjk,g^e=gjk,h^c=hj,h^d=hjk,i^a=ijk,i^b=ik,i^c=ik,i^d=ijk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338830, #72338906, #72339010, …
maximal normal subgroup of#72338830, #72338906, #72339010, …

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