Finite Group Databasebeta

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#72225472

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225471
commutatorC4×C32C22×C4
Frattini, cosocleC4×M64C23
prime radical#72216261C2
LCSG ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C8×M64 ◁ G
LECSG ▷ C4×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

571 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=fhk,d^2=ghik,e^2=k,f^2=hk,g^2=ijk,i^2=jk,j^2=k,b^a=bd,c^a=cik,c^b=cf,d^a=dg,d^b=dgh,e^b=eh,e^d=ek,f^a=fj,f^b=fhk,g^a=gi,g^b=gik,h^a=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225481, #72225510, #72225534
maximal normal subgroup of#72225481, #72225510, #72225534

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