Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24248 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C24×D8 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20385 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,14)(7,18)(9,26,23,16)(10,27,24,15)(11,12,25,17)(13,19)(20,30)(21,31)(22,34)(28,35)(29,33)(32,36);(6,12)(7,19)(9,10)(11,25)(13,31)(14,17)(15,27)(18,35)(20,21)(22,36)(23,24)(28,29)(30,34)(32,33);(1,3,4,2)(5,7,8,22)(6,13,14,32)(9,19,23,36)(10,28,24,20)(11,29,25,21)(12,30,17,35)(15,34,27,18)(16,31,26,33);(2,3)(13,32)(18,34)(30,35)(31,33)Polycyclic presentation:
2^10|a^2=f,c^2=fgh,g^2=i,h^2=ij,i^2=j,b^a=bi,c^a=cfgij,c^b=ch,d^a=dgh,d^c=de,e^a=ej,f^c=fj,g^a=gi,g^b=gij,g^c=gi,h^a=hij,h^b=hi,h^c=hi,i^a=ij,i^b=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72252664, #72252732, #72252954, … |
| maximal normal subgroup of | #72252664, #72252732, #72252954, … |
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.7*F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, 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);