Finite Group Databasebeta

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#72266645

order2048
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC22×C4#2840
commutator, Frattini, cosocleC25×C4C24
socleC23#16297
prime radical#7629120C22
LCSG ▷ C25×C4 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C25×C4 ▷ C23 ▷ C1

8329 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 8329 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 256

Polycyclic presentation:

2^11|a^2=e,b^2=efg,c^2=efjk,d^2=efij,h^2=k,b^a=bef,c^a=cfhk,c^b=cgjk,d^a=dh,d^b=dfgh,d^c=defk,e^b=ej,e^c=ei,e^d=eij,f^a=fj,f^b=fik,f^c=fi,f^d=fij,g^a=gik,g^b=gijk,g^c=gijk,g^d=gjk,h^a=hijk,h^c=hijk,h^d=hjk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72266655
Fitting subgroup of#72266655
maximal subgroup of#72266655
maximal normal subgroup of#72266655
maximal quotient of#72266649

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