Finite Group Databasebeta

← Prev (#72338783) Next (#72338785) →

#72338784

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72262365
commutatorC2×#379C22×C4
Frattini, cosocleC2×#2846C23
prime radical#6499867C22
LCSG ▷ C2×#379 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2846 ◁ G
LECSG ▷ C2×#2846 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4293 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 4293 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=f,c^2=ik,e^2=ik,f^2=k,b^a=bgi,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dk,e^a=eg,e^b=eik,e^c=eik,e^d=ejk,f^b=fik,f^c=fh,f^d=fjk,f^e=fi,g^a=gi,g^c=gj,g^e=gk,g^f=gk,h^a=hjk,h^b=hi,i^a=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338790, #72338810, #72338818
maximal normal subgroup of#72338790, #72338810, #72338818
maximal quotient of#72338987, #72338999, #72339001, …

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