Finite Group Databasebeta

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#72226918

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#123285
commutatorC8×C16C22×C4
Frattini, cosocleC2×C8×C16C23
socleC2#72225182
prime radical#72225394C2
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ G
LECSG ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

1656 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1656 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,8)(4,20)(5,9)(6,14)(7,10)(11,42)(12,22)(13,23)(15,33)(16,35)(17,45)(18,43)(19,31)(21,52)(24,32)(25,39)(26,41)(27,36)(28,34)(29,46)(30,49)(37,54)(38,56)(40,55)(44,58)(47,63)(48,60)(50,62)(51,61)(53,64)(57,59);(1,3)(2,8)(4,15)(5,24)(6,16)(7,28)(9,32)(10,34)(11,43)(12,45)(13,31)(14,35)(17,22)(18,42)(19,23)(20,33)(21,53)(25,57)(26,47)(27,46)(29,36)(30,40)(37,61)(38,60)(39,59)(41,63)(44,62)(48,56)(49,55)(50,58)(51,54)(52,64);(2,9,10,36,12,37,40,11,14,39,41,44,20,52,13,38)(3,15,16,17)(5,25)(8,29,30,59,33,60,34,61,35,62,31,32,45,43,63,64)(18,50,48,46)(19,47,49,28)(21,54)(24,51,57,53)(27,58)(42,56)

Polycyclic presentation:

2^11|c^2=d,d^2=gik,e^2=g,f^2=hk,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ce,d^a=dh,d^b=dg,e^a=ehjk,e^b=egi,f^a=fhjk,f^b=fhjk,f^c=fk,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226924, #72226961
maximal normal subgroup of#72226924, #72226961

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