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#72285762

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72285085C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1855 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1855 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:

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

Polycyclic presentation:

2^11|a^2=gk,b^2=de,c^2=ghjk,d^2=gj,f^2=gik,b^a=bd,c^a=chij,c^b=cf,d^a=dg,d^b=dh,d^c=dhj,e^a=eh,e^b=eh,e^c=ehi,e^d=ei,f^a=fh,f^b=fgjk,f^c=fgjk,f^d=fij,f^e=fik,g^b=gj,g^c=gjk,g^e=gk,g^f=gk,h^a=hk,h^b=hi,h^c=hjk,h^d=hjk,h^f=hj,i^a=ij,i^b=ijk,i^c=ij,i^d=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

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