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#72306345

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144386
commutator, Frattini, cosocleC2×#535C23
prime radical#417805C2
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1942 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1942 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)(3,14,19,40,17,45,20,44)(4,21)(5,26,15,42)(6,10)(7,12)(8,18,34,27,35,52,33,46)(9,25)(11,30,31,28)(13,29)(16,48,50,43,22,54,51,61)(23,56,47,63)(24,39)(32,57,36,49,62,64,53,55)(37,59)(38,58,41,60);(1,3,15,46)(2,8,31,44)(4,22,47,64)(5,27,7,17)(6,20,28,52)(9,36,41,61)(10,33,42,45)(11,40,12,35)(13,34,26,14)(16,23,57,25)(18,29,19,30)(21,53,38,43)(24,50,58,49)(32,56,54,37)(39,62,63,48)(51,60,55,59);(1,4,7,25)(2,9,12,21)(3,16,17,22)(5,23,15,47)(6,24,29,59)(8,32,35,62)(10,37,13,39)(11,41,31,38)(14,43,45,61)(18,49,52,55)(19,51,20,50)(26,56,42,63)(27,57,46,64)(28,58,30,60)(33,36,34,53)(40,48,44,54)

Polycyclic presentation:

2^10|a^2=f,b^2=efij,c^2=j,e^2=j,f^2=ij,h^2=j,b^a=be,c^a=cfj,c^b=cd,d^a=dh,d^b=dg,e^a=eghj,e^b=eghj,e^c=egij,e^d=ei,f^b=fghj,f^c=fij,f^d=fij,f^e=fi,g^a=gj,g^b=gj,h^a=hi,h^b=hi,h^c=hij,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306348, #72306358, #72306394, …
maximal normal subgroup of#72306348, #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.5*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.10);;
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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);