Finite Group Databasebeta

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#72306353

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144414
commutator, Frattini, cosocleC2×#535C23
prime radical#60644C22
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1286 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,14)(4,19,24,40,22,56,25,55)(5,26,28,38)(6,10)(7,12)(8,18)(9,23,36,27,37,59,42,60)(11,29,41,21)(13,30)(15,44,47,64,20,57,48,54)(16,49,51,62)(17,34)(31,58,35,61,43,46,63,50)(32,53)(33,45,39,52);(1,3,7,18)(2,8,12,14)(4,20,22,15)(5,16,28,51)(6,17,30,53)(9,35,37,63)(10,32,13,34)(11,39,41,33)(19,54,56,64)(21,45,29,52)(23,46,59,58)(24,47,25,48)(26,49,38,62)(27,61,60,50)(31,42,43,36)(40,57,55,44);(1,4,21,59)(2,9,26,19)(3,15,45,58)(5,27,30,25)(6,24,28,60)(7,22,29,23)(8,31,62,57)(10,36,11,40)(12,37,38,56)(13,42,41,55)(14,43,49,44)(16,50,53,48)(17,47,51,61)(18,20,52,46)(32,35,33,54)(34,63,39,64)

Polycyclic presentation:

2^10|a^2=f,b^2=j,c^2=efh,e^2=j,f^2=ij,h^2=j,b^a=bfj,c^a=ce,c^b=cd,d^a=dh,d^c=dg,e^a=eghij,e^b=eg,e^c=eghj,e^d=eij,f^b=fij,f^c=fghij,f^d=fij,f^e=fi,g^a=gj,g^c=gj,h^a=hi,h^b=hij,h^c=hij,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306355, #72306358, #72306394, …
maximal normal subgroup of#72306355, #72306358, #72306394, …

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