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#72225504

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225503
commutatorC4×C32C22×C4
Frattini, cosocleC4×M64C23
prime radical#513082C22
LCSG ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×#346 ◁ C4×#1398 ◁ G
LECSG ▷ C4×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

364 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^11|a^2=f,c^2=e,d^2=ghi,f^2=h,g^2=ij,h^2=k,i^2=jk,j^2=k,b^a=bd,c^a=cj,c^b=cfij,d^a=dg,d^b=dghk,d^c=djk,e^a=ek,e^b=ehj,e^d=ek,f^b=fhk,g^a=gi,g^b=gi,g^c=gk,h^b=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225510
maximal normal subgroup of#72225510

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