Finite Group Databasebeta

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#72343338

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#137479
commutatorC2×M32C22×C4
Frattini, cosocleC4×M32C23
prime radical#513022C2
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11404 ◁ G
LECSG ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

429 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=gi,d^2=g,e^2=j,f^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bhij,c^a=ce,c^b=cefh,d^b=di,e^a=ej,e^b=egij,e^c=ej,f^a=fgj,f^b=fghi,f^c=fhj,f^d=fj,f^e=fj,g^b=gj,h^a=hj,h^b=hi,h^c=hi,i^b=ij,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72343339, #72343345, #72343434, …
maximal normal subgroup of#72343339, #72343345, #72343434, …

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