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 | #7623112 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17336 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2307 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)(3,7)(4,12)(5,8)(6,9)(10,14)(11,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,38)(28,39)(29,40)(30,35)(31,41)(32,42)(33,37)(34,36)(43,48)(44,46)(45,47);(19,27)(20,32)(21,33)(24,38)(25,42)(26,37)(28,43)(29,34)(30,44)(31,45)(35,46)(36,40)(39,48)(41,47);(3,10)(7,14)(11,13)(15,16)(18,23)(24,35)(25,36)(26,37)(38,46)(39,47)(40,42)(41,48);(1,3,5,11,6,10,4,13)(2,7,8,16,9,14,12,15)(19,28,30,45)(20,29)(21,33)(24,39,35,47)(25,40)(26,37)(27,43,44,31)(32,34)(36,42)(38,48,46,41);(1,4,6,5)(2,8,9,12)(3,11,10,13)(7,15,14,16)(17,19,29,45,21,31,20,27,22,30,32,28,33,43,34,44)(18,24,40,47,26,41,25,38,23,35,42,39,37,48,36,46)Polycyclic presentation:
2^10|d^2=f,e^2=ghij,f^2=h,g^2=ij,i^2=j,c^a=cj,d^c=dh,e^a=eh,e^b=eg,e^d=eghi,f^e=fj,g^b=gi,g^d=gij,i^b=ij,i^d=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72287278, #72287431, #72287517, … |
| maximal normal subgroup of | #72287278, #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, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.9*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.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);