Finite Group Databasebeta

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#72338280

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#2196
commutator, Frattini, cosocle#390C24
LCSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

4518 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 4518 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,13,17)(11,14,18,23)(12,15,20,16)(19,22,21,24);(1,2)(3,5)(4,7)(6,8)(12,20)(14,22)(23,24);(2,5)(3,8)(6,7)(9,11,12,21,13,18,20,19)(10,14,15,24,17,23,16,22);(11,18)(14,23)(19,21)(22,24)

Polycyclic presentation:

2^10|a^2=j,c^2=gij,e^2=h,g^2=j,b^a=bf,c^b=ce,d^c=dj,e^a=eg,e^b=eh,e^c=ehi,f^c=fgij,f^e=fij,g^a=gj,g^b=gi,g^c=gj,g^e=gj,g^f=gj,i^a=ij,i^c=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338282
maximal normal subgroup of#72338282
maximal quotient of#72338412, #72339798

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