Finite Group Databasebeta

← Prev (#72341134) Next (#72341136) →

#72341135

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20585C23
prime radical#72284399C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1897 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1897 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,10)(3,12,15,40)(4,18,20,29)(6,11,24,32)(7,13,26,39)(8,28,30,57)(9,34,35,49)(14,41,46,60)(16,42,48,33)(17,45,25,31)(19,36,52,61)(21,44,53,59)(22,51,23,55)(27,43,58,62)(37,56,64,54)(38,63,50,47);(1,3)(2,8,13,33)(4,19,22,14)(5,15,26,25)(6,16)(7,17)(9,29,38,55)(10,30)(11,40,43,31)(12,32)(18,50)(20,52)(21,49,54,47)(23,46)(24,48,58,57)(27,28)(34,53)(35,51)(36,64,41,59)(37,60)(39,42)(44,61)(45,62)(56,63);(1,4)(2,9,13,38)(3,14)(5,23,26,20)(6,21,27,54)(7,22)(8,29)(10,35)(11,41)(12,44)(15,46,25,52)(16,49,28,47)(17,19)(18,42,51,30)(24,53)(31,59,40,64)(32,61,62,60)(33,55)(34,57)(36,43)(37,45)(39,50)(48,63)(56,58)

Polycyclic presentation:

2^11|a^2=d,b^2=ghij,c^2=hjk,e^2=k,f^2=gijk,b^a=bdek,c^a=cefgijk,c^b=chi,d^b=dij,d^c=dg,e^a=eijk,e^b=ej,e^c=egh,f^a=fik,f^b=fhk,f^c=fhij,f^d=fk,f^e=fi,g^a=gk,g^b=gjk,g^c=gk,g^e=gk,g^f=gk,h^a=hi,h^b=hj,h^d=hk,h^e=hk,h^f=hj,i^a=ik,i^b=ik,i^c=ij,i^f=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341156
maximal normal subgroup of#72341156

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