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#72225590

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2D162
commutator, Frattini, cosocleC8×C16C24
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G
LECSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

872 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^11|a^2=k,c^2=eg,d^2=fhk,e^2=gi,f^2=hj,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ce,d^a=dfk,d^b=df,e^a=eh,e^b=eg,f^a=fh,f^b=fh,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

185 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225593, #72227081, #72227082
maximal normal subgroup of#72225593, #72227081, #72227082

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