Finite Group Databasebeta

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#72285860

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216457
commutator, Frattini, cosocle#1781C24
prime radical#72210453C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3652 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 3652 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=gk,c^2=fghij,d^2=hk,f^2=hi,b^a=bgj,c^a=ce,c^b=cf,d^a=dgijk,d^b=dgij,d^c=dghj,e^b=eg,e^c=egk,e^d=ei,f^a=fgj,f^b=fh,f^c=fi,f^d=fi,f^e=fik,g^c=gi,g^d=gj,g^f=gjk,h^a=hjk,h^b=hjk,h^c=hj,h^e=hk,h^f=hk,i^b=ij,i^c=ijk,i^d=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286327
maximal normal subgroup of#72286327

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