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 | #7624074 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17330 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,4,8,13)(3,7,11,14,10,12,15,16)(17,18)(19,27,28,45)(20,33)(21,38)(22,24)(23,31,40,46)(25,36)(26,29)(30,44,34,42)(32,47,39,41)(35,43)(37,48);(1,3)(2,7)(4,12)(5,11)(6,10)(8,15)(9,14)(13,16)(19,28)(23,40)(27,45)(30,34)(31,46)(32,39)(41,47)(42,44);(1,4,5,13,6,2,8,9)(3,7,11,14,10,12,15,16)(17,19,29,32,22,28,21,39)(18,23,38,42,24,40,26,44)(20,34,35,46,37,30,36,31)(25,41,48,45,43,47,33,27);(2,4)(3,10)(9,13)(11,15)(18,24)(19,28)(25,43)(26,38)(30,34)(31,46)(32,39)(33,48);(3,10)(7,12)(11,15)(14,16)(18,24)(19,30)(20,35)(21,29)(23,41)(25,33)(27,42)(28,34)(31,39)(32,46)(36,37)(40,47)(43,48)(44,45)Polycyclic presentation:
2^10|a^2=f,c^2=fij,f^2=h,g^2=i,i^2=j,c^a=cg,c^b=chj,d^a=dhj,d^b=dh,d^c=dhj,e^a=ej,e^b=eh,e^c=eg,f^c=fj,g^a=gi,g^c=gi,g^e=gij,i^a=ij,i^c=ij,i^e=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72287684, #72318841, #72318999, … |
| maximal normal subgroup of | #72287684, #72318841, #72318999, … |
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.6);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
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.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);