Finite Group Databasebeta

← Prev (#72252411) Next (#72252413) →

#72252412

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72210489
commutator#2309C42
Frattini, prime radical, cosocle#6628901C22
LCSG ▷ #2309 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6628901 ◁ G
LECSG ▷ #6628901 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1

5597 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 5597 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=e,d^2=ij,e^2=k,f^2=k,b^a=bcdij,c^b=cg,d^a=dghi,d^b=df,d^c=dhk,e^a=efghij,e^d=eik,f^a=fh,f^b=fi,f^c=fk,f^d=fk,f^e=fk,g^a=ghj,g^d=gj,g^f=gk,h^a=hk,h^b=hj,h^d=hk,h^e=hk,i^a=ij,i^b=ik,i^c=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252413, #72267057, #72267059, …
maximal normal subgroup of#72252413, #72267057, #72267059, …

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