Finite Group Databasebeta

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#72238385

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72236911
commutator, Frattini, cosocleC2×#2613C23
prime radical#72237377C2
LCSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

3196 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 3196 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,13)(4,8,15,23)(5,10)(6,11,21,30)(7,12)(9,17,26,32)(14,24)(16,29)(18,22,19,27)(20,28)(25,31);(1,3)(2,8)(4,6)(5,14)(7,16)(9,10)(11,29)(12,23)(13,30)(15,21)(17,19)(18,32)(20,31)(22,25)(24,27)(26,28);(1,4,6,3)(2,9,11,25,12,26,30,24)(5,17,19,14)(7,15,21,16)(8,22,13,10,23,27,29,28)(18,31,20,32)

Polycyclic presentation:

2^11|a^2=ghi,c^2=f,e^2=k,f^2=ij,b^a=be,c^a=cd,c^b=cf,d^a=dj,d^b=dg,d^c=dgh,e^a=eijk,e^b=ek,e^c=ehijk,e^d=ej,f^a=fgh,f^b=fij,f^d=fjk,f^e=fijk,g^a=gj,g^c=gi,g^e=gk,g^f=gk,h^a=hjk,h^b=hijk,h^c=hijk,i^a=ik,i^b=ik,i^c=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238387, #72262498, #72262580
maximal normal subgroup of#72238387, #72262498, #72262580

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