Finite Group Databasebeta

← Prev (#72266428) Next (#72266430) →

#72266429

order2048
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C212

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC24#2840
commutator, Frattini, cosocleC27C24
prime radicalC24×#120C22
LCSG ▷ C27 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C27 ◁ G
LECSG ▷ C27 ▷ C22 ▷ C1

81030 conjugacy classes of subgroups; only the 740 normal subgroups are shown.

normal subgroups only

This group has 81030 conjugacy classes of subgroups — too many to draw. Only its 740 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) = 128

Polycyclic presentation:

2^11|a^2=e,b^2=efg,c^2=ei,d^2=egik,b^a=bef,c^a=ci,c^b=cgh,d^a=dhijk,d^b=dfik,d^c=defgijk,e^b=ej,e^c=ejk,f^a=fj,f^b=fjk,f^c=fj,f^d=fjk,g^a=gjk,g^b=gk,g^c=gjk,g^d=gk,h^a=hjk,h^b=hjk,h^d=hk,i^a=ijk,i^b=ik,i^c=ijk,i^d=ik

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72266433
Fitting subgroup of#72266433
maximal subgroup of#72266433
maximal normal subgroup of#72266433
maximal quotient of#72266483

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