Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13086 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| socle | C2 | C2×#13086 |
| prime radical | #6437444 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C24×C4 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
4105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8,16,28)(4,9)(5,21,22,10)(6,13,25,12)(7,11)(14,26,39,53)(15,41,43,27)(17,29)(18,46,48,33)(19,31)(20,32,49,36)(23,37,24,34)(30,47,58,50)(35,45)(38,59,60,56)(40,55)(42,51,44,57)(52,62,64,61)(54,63);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,26)(10,27)(11,28)(12,35)(13,29)(18,38)(19,39)(20,40)(21,51)(22,44)(23,42)(24,43)(25,45)(30,52)(31,53)(32,55)(33,56)(34,57)(36,54)(37,41)(46,64)(47,62)(48,61)(49,63)(50,60)(58,59);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,26)(10,33)(11,31)(12,36)(13,32)(15,38)(16,39)(17,40)(21,46)(22,48)(23,47)(24,50)(25,49)(27,56)(28,53)(29,55)(30,34)(35,54)(37,58)(41,59)(42,62)(43,60)(44,61)(45,63)(51,64)(52,57);(1,5)(2,10)(3,15)(4,18)(6,24)(7,23)(8,27)(9,30)(11,34)(12,21)(13,37)(14,38)(16,42)(17,43)(19,47)(20,50)(22,25)(26,52)(28,57)(29,41)(31,33)(32,46)(35,51)(36,58)(39,62)(40,60)(44,45)(48,49)(53,56)(54,59)(55,64)(61,63)Polycyclic presentation:
2^10|a^2=g,h^2=j,b^a=bi,d^a=df,d^c=de,e^a=ej,f^a=fh,f^b=fj,f^c=fj,g^b=gj,g^d=ghj,g^f=gij,h^a=hi,h^d=hj,h^f=hj,h^g=hj,i^a=ij,i^d=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72284492, #72286823, #72286893, … |
| maximal normal subgroup of | #72284492, #72286823, #72286893, … |
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.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, 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, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);