Finite Group Databasebeta

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#72225490

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225489
commutatorC4×C32C42
Frattini, cosocleC8×M64C22
prime radicalC23#7453
LCSG ▷ C4×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C8×M64 ◁ G
LECSG ▷ C8×M64 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

237 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=degijk,c^2=fghj,d^2=gik,e^2=gj,f^2=hjk,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bc,c^a=cf,c^b=cfgi,d^b=dgij,d^c=dk,e^a=ej,e^b=egij,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225495, #72225515, #72225523
maximal normal subgroup of#72225495, #72225515, #72225523

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