Finite Group Databasebeta

← Prev (#72266698) Next (#72266700) →

#72266699

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8535
commutatorC23×D4C22×C4
Frattini, cosocleC22×#120C23
prime radical#15947C22
LCSG ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G
LECSG ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1

2045 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 2045 conjugacy classes of subgroups — too many to draw. Only its 87 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)(3,11,13,7)(4,16,18,8)(5,10)(6,9)(12,27,14,26)(15,21,29,23)(17,22,30,32)(19,31,28,24)(20,25);(1,3)(2,7)(4,17,19,14)(5,15)(6,13)(8,22,24,26)(9,11)(10,23)(12,28,30,18)(16,27,31,32)(20,29)(21,25);(2,8)(3,12)(7,21)(9,24)(10,16)(11,23)(13,30)(14,29)(15,17)(22,32)(25,31)(26,27)

Polycyclic presentation:

2^10|a^2=d,b^2=fh,b^a=bdf,c^a=ce,c^b=ceij,d^b=dj,d^c=dg,e^a=eg,e^b=eh,e^d=ei,f^a=fj,f^b=fj,f^c=fh,f^e=fj,g^a=gi,g^b=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266700, #72266711, #72266715
maximal normal subgroup of#72266700, #72266711, #72266715
maximal quotient of#72337727

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