Finite Group Databasebeta

← Prev (#72262805) Next (#72262807) →

#72262806

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23984
commutator, Frattini, cosocleC22×C22⋊C4C24
prime radical#62817C22
LCSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

2637 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 2637 conjugacy classes of subgroups — too many to draw. Only its 137 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gij,e^2=h,b^a=bgij,c^a=ce,c^b=cf,d^a=dgh,d^b=dj,d^c=dh,e^a=ej,e^b=ei,e^c=eh,e^d=ehj,f^a=fhij,f^b=fh,f^e=fj,g^c=ghj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72262808, #72262833
maximal normal subgroup of#72262808, #72262833

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