Finite Group Databasebeta

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#72227264

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#9572
commutatorC82C22×C4
Frattini, cosocleC2×C82C23
prime radical#19603C22
LCSG ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G
LECSG ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1

366 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 256

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

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

Polycyclic presentation:

2^10|a^2=d,b^2=dfhij,c^2=dfg,d^2=hij,e^2=gh,f^2=h,g^2=i,h^2=j,b^a=bf,c^a=ce,c^b=cfgj,d^b=dh,d^c=dhij,e^a=eg,e^b=ehij,e^c=egh,f^a=fj,f^b=fh,f^c=fh,g^a=gi,g^c=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227286
maximal normal subgroup of#72227286
maximal quotient of#72227266

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