Finite Group Databasebeta

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#72283512

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#13673
commutator#493C22×C4
Frattini, cosocle#3147C23
socleC2#152185
prime radical#6358420C2
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1079 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1079 conjugacy classes of subgroups — too many to draw. Only its 75 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:

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

Polycyclic presentation:

2^10|a^2=ghi,b^2=i,c^2=dfj,f^2=j,g^2=i,h^2=j,b^a=bi,c^a=cd,c^b=ce,d^a=dhj,d^b=dg,e^a=eghi,e^b=ej,e^d=ehij,f^a=fhj,f^b=fg,f^e=fhi,g^a=gi,g^b=gi,g^c=gh,g^d=gi,g^e=gi,g^f=gi,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283540, #72284059, #72340154
maximal normal subgroup of#72283540, #72284059, #72340154

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