Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7624079 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17320 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,30)(20,26)(21,38)(22,28)(23,25)(24,29)(27,34)(31,46)(32,42)(33,48)(35,43)(36,44)(37,41)(39,45)(40,47);(19,23)(20,21)(22,35)(25,30)(26,38)(27,44)(28,43)(31,40)(32,37)(33,39)(34,36)(41,42)(45,48)(46,47);(1,3,5,12,6,13,4,14)(2,7,9,15,10,11,8,16)(17,19)(18,25)(20,34,36,21)(22,39)(23,24)(26,38,44,27)(29,30)(31,32,37,40)(41,42,46,47)(43,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,35,40,23,37,39,34,24,36,33,32,19,31,22,21)(18,26,43,47,25,41,45,27,29,44,48,42,30,46,28,38);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,21,39,37,19,32,35,20,24,34,22,31,23,40,33,36)(18,27,45,46,30,47,43,44,29,38,28,41,25,42,48,26)Polycyclic presentation:
2^10|c^2=f,d^2=ghi,e^2=ghj,f^2=h,g^2=ij,i^2=j,c^a=chj,c^b=cj,d^b=dg,d^c=dghij,e^a=ehj,e^b=egi,e^c=eghj,e^d=ej,f^d=fj,f^e=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72287030, #72287214, #72287517, … |
| maximal normal subgroup of | #72287030, #72287214, #72287517, … |
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, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);