Finite Group Databasebeta

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#72321395

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130718
commutatorC22×C16C22×C4
Frattini, cosocleC2×#504C23
prime radical#7623634C2
LCSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33435 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

1152 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 115 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=dfgj,d^2=hi,e^2=hi,f^2=j,h^2=ij,i^2=j,b^a=bd,c^a=cdefh,c^b=cdgh,d^a=dg,d^b=dh,d^c=dgj,e^b=egh,e^c=egj,e^d=ej,f^b=fg,f^c=fg,f^d=fj,g^a=gj,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#72321495, #72342911, #72342960
maximal normal subgroup of#72321495, #72342911, #72342960

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