Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #21863 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| socle | C2 | #382487 |
| prime radical | #10575347 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#2875 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,10)(4,21,25,27)(5,6)(11,34)(12,39,35,42)(15,33)(16,44,47,23)(17,32)(22,52,41,58)(24,54,26,53)(29,30)(36,62,61,40)(38,51)(45,63,64,55)(46,56,60,57)(49,50);(1,2)(3,11)(4,22,25,41)(5,28)(6,31)(7,15)(8,33)(9,10)(12,26,35,24)(13,30)(14,29)(16,45,47,64)(17,48)(18,38)(19,51)(20,34)(21,52,27,58)(23,55,44,63)(32,59)(36,46,61,60)(37,50)(39,53,42,54)(40,57,62,56)(43,49);(1,3)(2,11)(4,23,25,44)(5,17)(6,32)(7,18)(8,19)(9,20)(10,34)(12,40,35,62)(13,37)(14,43)(15,38)(16,27,47,21)(22,55,41,63)(24,56,26,57)(28,48)(29,49)(30,50)(31,59)(33,51)(36,42,61,39)(45,58,64,52)(46,54,60,53);(1,4,5,24,8,27,30,54,9,25,6,26,7,21,29,53)(2,12,13,41,15,42,28,58,10,35,14,22,33,39,31,52)(3,16,17,46,19,23,50,56,20,47,32,60,18,44,49,57)(11,36,37,45,38,40,48,63,34,61,43,64,51,62,59,55)Polycyclic presentation:
2^10|a^2=f,b^2=f,c^2=f,d^2=e,e^2=i,g^2=i,h^2=j,i^2=j,b^a=bfj,d^a=dh,d^b=dg,d^c=dfij,e^a=ej,e^b=ei,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72308411 |
| maximal normal subgroup of | #72308411 |
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, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);