Finite Group Databasebeta

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#72227557

order2048
nilpotency class7
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227526
commutator, Frattini, cosocle#8546C23
prime radicalC2×#2867D4
LCSG ▷ #8546 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C25 ◁ C23×D4 ◁ #8546 ◁ G
LECSG ▷ #8546 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1002 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 40 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:

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

Polycyclic presentation:

2^11|a^2=fhijk,b^2=ghjk,c^2=gh,d^2=fijk,e^2=fhijk,b^a=bd,c^a=ce,c^b=cghik,d^a=df,d^b=dfhik,d^c=dg,e^a=efhi,e^b=egj,e^c=efik,e^d=eh,f^a=fjk,f^b=fj,f^c=fik,f^d=fk,f^e=fj,g^a=gh,g^b=gik,g^c=gij,g^d=gik,g^e=gijk,g^f=gk,h^a=hk,h^b=hi,h^c=hij,h^d=hjk,h^e=hjk,i^a=ij,i^c=ik,i^d=ik,i^e=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227558, #72227562

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