Finite Group Databasebeta

← Prev (#72341066) Next (#72341068) →

#72341067

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#1304
commutatorC25C2×C42
Frattini, cosocleC22×#127C23
prime radicalC23×#127C22
LCSG ▷ C25 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×#127 ◁ G
LECSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1

4097 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 4097 conjugacy classes of subgroups — too many to draw. Only its 161 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) = 64

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

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

Polycyclic presentation:

2^10|a^2=d,c^2=f,d^2=i,b^a=bhj,c^a=ce,c^b=cij,d^b=di,d^c=dg,e^a=eg,e^b=eij,e^c=eh,e^d=ei,f^a=fh,f^d=fi,f^e=fij,g^a=gi,g^c=gi,h^a=hi,h^c=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341069, #72341191, #72341208
maximal normal subgroup of#72341069, #72341191, #72341208

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