Finite Group Databasebeta

← Prev (#72311225) Next (#72311227) →

#72311226

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216457
commutator, Frattini, cosocle#1781C24
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3564 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 3564 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=ghj,c^2=ehij,d^2=k,e^2=h,b^a=bghj,c^a=ce,c^b=cf,d^a=dhj,d^b=di,d^c=dhj,e^a=eh,e^b=eg,e^c=ej,e^d=ek,f^a=fgj,f^b=fik,f^c=fgj,f^d=fk,f^e=fi,g^b=gjk,g^c=gi,g^d=gk,g^e=gjk,h^b=hjk,h^f=hk,i^a=ij,i^b=ik,i^c=ijk,i^e=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311339, #72340399, #72340466
maximal normal subgroup of#72311339, #72340399, #72340466

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