Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #147314 |
| commutator | C22×Q8 | C2×C42 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C23×D4 | C2×D4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#117 ◁ #34211 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,8)(4,17,18,48)(5,22,23,36)(6,25,26,11)(7,12)(9,20,32,51)(10,24)(13,37,38,59)(14,29)(15,44,42,58)(16,46,43,31)(19,49,21,50)(27,35)(28,40,54,62)(30,45)(33,52,34,53)(39,60,41,61)(47,57)(55,63,56,64);(1,3)(2,8)(4,13,19,39)(5,15)(6,16)(7,14)(9,28,33,55)(10,30)(11,31)(12,29)(17,37,49,60)(18,38,21,41)(20,40,52,63)(22,44)(23,42)(24,45)(25,46)(26,43)(27,47)(32,54,34,56)(35,57)(36,58)(48,59,50,61)(51,62,53,64);(1,4)(2,9,12,33)(3,13,14,39)(5,20)(6,18)(7,19)(8,28)(10,17,35,49)(11,32,25,34)(15,40,42,63)(16,38,43,41)(21,26)(22,48,36,50)(23,52)(24,53)(27,51)(29,55)(30,37)(31,54)(44,59)(45,64,47,62)(46,56)(57,60)(58,61)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=ei,g^2=j,h^2=j,c^a=cg,c^b=ce,d^c=dh,e^b=ej,e^c=ej,f^c=fj,g^a=gh,g^d=gij,h^a=hi,h^c=hj,h^d=hj,h^g=hj,i^a=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317866, #72318077, #72318097, … |
| maximal normal subgroup of | #72317866, #72318077, #72318097, … |
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.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);