Finite Group Databasebeta

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#72314005

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8541
commutatorC22×C22⋊C4C22×C4
Frattini, cosocleC23×C22⋊C4C23
prime radical#6512855C2
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C23×C22⋊C4 ▷ C24 ▷ C2 ▷ C1

2647 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 2647 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,d^2=i,b^a=bd,c^b=cf,d^a=di,d^b=dg,d^c=di,e^a=egi,e^c=eh,f^a=fij,f^b=fh,f^d=fj,g^c=gj,h^a=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72314021, #72316377
maximal normal subgroup of#72314021, #72316377

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