Finite Group Databasebeta

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#72227524

order1024
nilpotency class7
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#94703
commutator#1781C2×C4
Frattini, cosocle#8541C22
prime radicalC22×#120D4
LCSG ▷ #1781 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C25 ◁ C23×D4 ◁ #8541 ◁ G
LECSG ▷ #8541 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1

494 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 23 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=fhi,c^2=eg,d^2=i,f^2=j,b^a=bc,c^a=cf,c^b=ce,d^b=defg,d^c=dgj,e^a=ehj,e^b=ehi,e^c=ei,e^d=ei,f^a=fg,f^b=fj,f^c=fhij,f^d=fij,f^e=fj,g^a=gij,g^b=gh,g^c=gi,h^a=hi,h^c=hj,i^b=ij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227531
maximal normal subgroup of#72227531

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