Finite Group Databasebeta

← Prev (#72284673) Next (#72284675) →

#72284674

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237816
commutator#2308C22×C4
Frattini, cosocle#20585C23
prime radical#72284642C2
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432954 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3430 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3430 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^11|b^2=e,c^2=gj,d^2=h,f^2=hijk,b^a=bd,c^a=cg,c^b=cefhijk,d^a=dh,d^b=dg,d^c=dik,e^a=eghjk,e^c=egh,e^d=ei,f^a=fghk,f^b=fg,f^c=fgijk,f^d=fij,f^e=fi,g^b=gi,g^d=gjk,g^e=gk,g^f=gjk,h^b=hjk,h^c=hk,h^e=hk,h^f=hk,i^a=ij,i^b=ik,i^c=ijk,i^d=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286327
maximal normal subgroup of#72286327

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