Finite Group Databasebeta

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#72339876

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152191
commutator#492C22×C4
Frattini, cosocle#3151C23
prime radical#6365636C2
LCSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G
LECSG ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1049 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1049 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,9)(4,15,20,32,21,30,19,28)(5,22)(6,7)(8,12)(10,17,18,16,27,31,13,23)(11,26)(14,24)(25,29);(1,3)(2,9)(4,16,20,31,21,23,19,17)(5,11)(6,8)(7,12)(10,28,18,15,27,32,13,30)(14,24)(22,26)(25,29);(1,4,5,23,9,19,25,31,7,21,26,16,8,20,24,17)(2,10,22,32,3,13,29,15,6,27,11,28,12,18,14,30)

Polycyclic presentation:

2^10|a^2=gh,b^2=gh,c^2=dfij,d^2=hj,e^2=j,f^2=ij,g^2=i,h^2=j,c^a=cd,c^b=cdej,d^a=dj,d^b=dg,d^c=dhij,e^a=eghj,e^b=egh,e^c=egh,e^d=eh,f^a=fi,f^b=fj,f^c=fhij,f^d=fj,f^e=fi,g^b=gi,g^c=gh,g^e=gi,h^b=hi,h^c=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339877, #72339889
maximal normal subgroup of#72339877, #72339889

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