Finite Group Databasebeta

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#72284550

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139505
commutatorC22×C22⋊C4C22×C4
Frattini, cosocle#2308C23
prime radical#23444C22
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35448 ◁ G
LECSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1126 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1126 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=f,c^2=j,e^2=gi,b^a=bgj,c^a=cd,c^b=ce,d^a=dj,d^b=dg,e^a=eg,e^b=egh,e^c=egi,e^d=ej,f^a=fj,f^c=fhi,f^d=fj,f^e=fij,g^b=gj,g^c=gj,h^a=hj,h^b=hi,h^c=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284595, #72284615, #72285575
maximal normal subgroup of#72284595, #72284615, #72285575

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