Finite Group Databasebeta

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#72340932

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139622
commutator#379C22×C4
Frattini, cosocle#2308C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G
LECSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1652 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1652 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=ghj,e^2=gij,b^a=bdegij,c^a=chj,c^b=cfghij,d^b=dg,e^a=ei,e^b=eh,e^c=ehi,e^d=ej,f^a=fhi,f^c=fij,f^d=fj,f^e=fi,g^a=gj,g^b=gj,g^c=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340933, #72340946
maximal normal subgroup of#72340933, #72340946

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