Finite Group Databasebeta

← Prev (#72306640) Next (#72306642) →

#72306641

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13674
commutator, Frattini, cosocle#492C24
LCSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1436 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1436 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=ghi,c^2=ghij,e^2=hj,g^2=i,h^2=j,b^a=bij,c^b=ci,d^a=dj,d^b=df,d^c=de,e^b=eg,e^d=eh,f^a=fj,f^b=fhj,f^c=fghj,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306647
maximal normal subgroup of#72306647

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