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#72227058

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72216562
commutatorC8×C16C2×C42
Frattini, cosocleC2×C162C23
prime radicalC2×#123403C22
LCSG ▷ C8×C16 ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C82 ◁ #72216182 ◁ G
LECSG ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1

1657 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1657 conjugacy classes of subgroups — too many to draw. Only its 203 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) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=e,c^2=f,d^2=ghi,e^2=h,g^2=ik,h^2=jl,i^2=k,j^2=l,b^a=bd,c^a=cij,c^b=ckl,d^a=dg,d^b=dghj,d^c=dkl,e^b=ehj,e^c=el,f^a=fkl,g^a=gi,g^b=gi,h^b=hj,i^a=ik,i^b=ik,j^b=jl

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227169
maximal normal subgroup of#72227169
maximal quotient of#72227059

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.11*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);