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#72268178

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23900
commutator, Frattini, cosocleC23×Q8C24
prime radical#62833C22
LCSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

3693 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 3693 conjugacy classes of subgroups — too many to draw. Only its 167 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=g,d^2=i,e^2=j,f^2=j,b^a=bf,c^a=ce,c^b=chi,d^b=dgh,d^c=dgh,e^a=ej,e^b=eh,e^c=ei,e^d=ei,f^a=fj,f^b=fi,f^c=fh,f^d=fi,f^e=fj,g^a=gij,h^a=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268180
maximal normal subgroup of#72268180

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