Finite Group Databasebeta

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#72285726

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#251828C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1799 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1799 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=jk,c^2=gj,d^2=k,e^2=hik,f^2=hjk,b^a=bf,c^a=ce,c^b=cj,d^b=dh,d^c=dhi,e^a=ek,e^b=eg,e^c=ehk,f^a=fjk,f^b=fhj,f^c=fgij,f^d=fk,f^e=fijk,g^a=gi,g^d=gk,g^e=gj,g^f=gjk,h^a=hk,h^c=hj,h^e=hk,i^a=ik,i^b=ij,i^c=ijk,i^e=ik,i^f=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285850
maximal normal subgroup of#72285850

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