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 | |
2266 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(3,10)(6,9)(11,16)(13,15)(17,18)(19,25)(20,24)(21,26)(22,27)(23,39)(28,37)(29,41)(30,42)(31,35)(32,45)(33,38)(34,36)(40,43)(44,46)(47,48);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19,22,32,24,34,41,35)(18,25,27,45,20,36,29,31)(21,38,39,30,28,47,40,46)(23,42,37,48,43,44,26,33);(17,20,37,40,22,27,43,28,24,18,26,39,41,29,23,21)(19,30,44,36,35,47,48,31,34,46,42,25,32,38,33,45);(2,7)(6,9)(8,14)(13,15)(17,21,23,29,41,39,26,18,24,28,43,27,22,40,37,20)(19,31,33,47,32,36,42,30,34,45,48,38,35,25,44,46)Polycyclic presentation:
2^10|c^2=f,d^2=gi,e^2=g,f^2=h,g^2=ij,i^2=j,c^a=chj,c^b=chj,d^b=dg,d^c=dgi,e^a=ehj,e^b=egi,e^c=egj,f^d=fj,f^e=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij103 conjugacy classes — view the full character table.
| maximal subgroup of | #72287214, #72287431, #72287517, … |
| maximal normal subgroup of | #72287214, #72287431, #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.9);;
gap> SetPower(coll, 5, F.7);;
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.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
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.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);