Finite Group Databasebeta

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#72321082

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130798
commutatorC22×C16C22×C4
Frattini, cosocleC2×#504C23
prime radical#7623852C2
LCSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33435 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

792 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 111 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=defh,d^2=j,e^2=hi,f^2=i,h^2=ij,i^2=j,b^a=bg,c^a=cd,c^b=cdeh,d^c=dg,e^a=ej,e^b=eh,e^c=ej,f^a=fg,f^b=fgij,f^c=fgj,f^d=fj,g^c=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321146, #72321495, #72342984
maximal normal subgroup of#72321146, #72321495, #72342984

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