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#72262495

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227790
commutator, Frattini, cosocleC2×#2613C23
prime radical#72237376C2
LCSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

3437 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 3437 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,6)(4,9,20,28)(5,21)(7,14)(8,11)(10,25)(12,19,27,18)(13,16)(15,31,30,29)(17,32,23,26)(22,24);(1,3,6,2)(4,15,19,32,20,30,18,26)(5,22,24,21)(7,11,8,14)(9,17,12,29,28,23,27,31)(10,25,13,16);(1,4,16,29)(2,9,22,15)(3,12,21,32)(5,23,6,18)(7,19,25,17)(8,20,24,31)(10,26,14,27)(11,28,13,30)

Polycyclic presentation:

2^11|a^2=ijk,b^2=eijk,c^2=dek,d^2=ik,e^2=ijk,f^2=k,b^a=be,c^a=cefk,c^b=cd,d^a=dg,d^b=dgh,d^c=dghik,e^a=eijk,e^c=eghi,e^d=ej,f^a=fij,f^b=fhij,f^c=fhjk,f^d=fik,f^e=fij,g^b=gi,g^d=gk,g^e=gk,g^f=gk,h^a=hij,h^b=hij,h^c=hjk,i^a=ik,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#72262498, #72262580, #72262767
maximal normal subgroup of#72262498, #72262580, #72262767

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