Finite Group Databasebeta

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#72225457

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225456
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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=d,c^2=fghjk,d^2=ij,e^2=gj,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bc,c^a=cf,c^b=cfgi,d^a=djk,d^c=dk,e^b=egij,f^a=fh,f^b=fhk,g^a=gk,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#72225464, #72225523, #72225534
maximal normal subgroup of#72225464, #72225523, #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, 2, F.4);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
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, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
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*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
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);