Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8535 |
| commutator | C23×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#121 | C23 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1 | |
1302 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 1302 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.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,6)(3,9)(4,12,13,8)(7,11)(10,15)(14,16)(17,18)(19,23)(20,30,33,24)(21,25)(22,26)(27,38,31,35)(28,36)(29,37)(32,43,45,39)(34,40)(41,48,44,46)(42,47);(2,6)(3,10)(4,13)(7,15)(9,16)(11,14)(17,19,22,29)(18,23,26,37)(20,31,33,27)(21,28,34,42)(24,38,30,35)(25,36,40,47)(32,41,45,44)(39,46,43,48);(1,3)(2,7)(4,14)(5,11)(6,9)(8,15)(10,13)(12,16)(17,20,21,32,22,33,34,45)(18,24,25,39,26,30,40,43)(19,27,28,41,29,31,42,44)(23,35,36,46,37,38,47,48)Polycyclic presentation:
2^10|a^2=gh,b^2=j,c^2=d,d^2=j,e^2=i,b^a=bi,c^a=cf,c^b=ce,d^b=dg,e^a=eh,e^b=ei,e^c=egi,e^d=ej,f^a=fij,f^b=fhi,f^e=fi,g^c=gj,h^c=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72303984 |
| maximal normal subgroup of | #72303984 |
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.7*F.8);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);