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#72225589

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

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

normal subgroups only

This group has 898 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)(3,16,21,46)(4,8,24,30)(5,10,26,38)(6,12,25,15)(9,32,36,17)(11,40,41,18)(13,43,39,22)(19,44,55,59)(20,45,54,47)(23,28,51,31)(27,58,60,48)(29,57,42,52)(33,61,62,49)(34,56,37,50)(35,64,63,53);(3,17)(4,18)(5,22)(6,25)(8,11)(9,16)(10,13)(12,15)(19,53)(20,56)(21,32)(23,57)(24,40)(26,43)(27,33)(28,42)(29,31)(30,41)(34,47)(35,44)(36,46)(37,45)(38,39)(48,49)(50,54)(51,52)(55,64)(58,61)(59,63)(60,62);(2,8,13,29,12,28,38,41,14,30,39,42,15,31,10,11)(9,27,35,45,34,59,62,46,36,60,63,47,37,44,33,16);(1,3,18,49,5,19,51,56,25,54,57,64,43,58,24,32,7,21,40,61,26,55,23,50,6,20,52,53,22,48,4,17)(2,9,8,27,13,35,29,45,12,34,28,59,38,62,41,46,14,36,30,60,39,63,42,47,15,37,31,44,10,33,11,16)

Polycyclic presentation:

2^11|a^2=k,c^2=eg,d^2=fh,e^2=gi,f^2=hj,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ce,d^a=df,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, #72227076, #72227080
maximal normal subgroup of#72225593, #72227076, #72227080

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);;
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);;
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);