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#72268477

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23984
commutator, Frattini, cosocleC23×Q8C24
prime radical#62824C22
LCSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

2315 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 2315 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18,22,25)(19,28,29,43)(20,32,35,26)(21,27,36,42)(23,33,40,30)(24,34,41,37)(31,38,45,48)(39,47,44,46);(1,3)(2,6,8,9)(4,10)(5,11)(7,16,12,15)(13,14)(17,19)(18,23)(20,33)(21,31)(22,29)(24,38)(25,40)(26,43)(27,44)(28,32)(30,35)(34,47)(36,45)(37,46)(39,42)(41,48);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(18,24)(21,36)(23,38)(25,41)(26,42)(27,32)(28,44)(31,45)(34,37)(39,43)(40,48)(46,47);(1,3)(2,6)(4,13)(5,11)(7,16)(8,9)(10,14)(12,15)(18,25)(19,29)(21,36)(26,32)(30,33)(34,37)(38,48)(39,44)

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=g,e^2=j,f^2=j,b^a=be,c^a=cefj,c^b=cg,d^a=di,d^b=dgj,d^c=dh,e^a=ej,e^b=ei,e^c=eh,e^d=eij,f^b=fh,f^c=fhi,f^d=fi,f^e=fj,g^a=gij,h^a=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268486
maximal normal subgroup of#72268486

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