Finite Group Databasebeta

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#72285259

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72284401C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2127 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2127 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=jk,c^2=e,e^2=h,f^2=hijk,b^a=bjk,c^a=cd,c^b=cf,d^b=dg,d^c=dg,e^a=eg,e^b=eg,e^d=ei,f^a=fgi,f^b=fhij,f^c=fghi,f^d=fik,f^e=fij,g^c=gi,g^e=gj,g^f=gjk,h^a=hj,h^b=hj,h^d=hk,h^f=hk,i^b=ij,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285547, #72286183, #72340224, …
maximal normal subgroup of#72285547, #72286183, #72340224, …

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