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#72339612

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139700
commutatorC2×#127C22×C4
Frattini, cosocleC2×#535C23
prime radicalC2×#3019C22
LCSG ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1730 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1730 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,9)(4,20)(5,26)(6,13)(7,12)(8,14)(10,23)(11,30)(15,43)(16,47)(17,34)(18,33)(19,35)(21,36)(22,37)(24,39)(25,38)(27,32)(28,41)(29,42)(31,44)(40,49)(45,61)(46,60)(48,58)(50,52)(51,55)(53,64)(54,62)(56,59)(57,63);(1,3)(2,9)(4,21,24,53)(5,27,28,48)(6,18,7,17)(8,19)(10,36,38,64)(11,40)(12,33,13,34)(14,35)(15,22,45,56)(16,29)(20,50,39,62)(23,52,25,54)(26,58,41,32)(30,49)(31,37,60,59)(42,47)(43,51,61,63)(44,55,46,57);(1,4,6,24,8,25,7,23)(2,10,12,38,14,39,13,20)(3,15,17,44,19,46,18,45)(5,22,29,56,28,57,30,55)(9,31,33,43,35,61,34,60)(11,37,41,51,42,63,26,59)(16,21,27,53,49,54,48,52)(32,36,47,50,58,62,40,64)

Polycyclic presentation:

2^10|b^2=egh,c^2=f,d^2=i,f^2=j,g^2=j,h^2=j,b^a=be,c^a=cgj,c^b=cd,d^a=dg,d^b=dgi,d^c=dghj,e^c=eg,e^d=ei,f^a=fj,f^b=fghij,f^d=fj,f^e=fj,g^a=gj,g^b=gi,g^e=gj,h^a=hj,h^b=hi,h^c=hj,h^e=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339616
maximal normal subgroup of#72339616
maximal quotient of#72339622

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