Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #496225 |
| commutator | C22×Q8 | C23×C4 |
| Frattini, cosocle | C2×#124 | C24 |
| prime radical | #6437459 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ C2×#3499 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3521 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)(3,13,15,8)(4,18,21,53,22,54,19,38)(5,24,20,39)(6,12)(7,10)(9,23,32,55,35,56,34,25)(11,26,36,27)(14,42,43,63,44,60,51,61)(16,48)(17,41,47,31)(28,37)(29,45,59,49,33,62,58,64)(30,50)(40,52)(46,57);(4,19,22,21)(9,32,35,34)(14,43,44,51)(18,38,54,53)(23,55,56,25)(29,58,33,59)(42,63,60,61)(45,64,62,49);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,33)(10,13)(11,30)(12,31)(18,42)(19,43)(20,46)(21,51)(22,44)(23,45)(24,57)(25,49)(26,50)(27,52)(28,47)(29,35)(32,59)(34,58)(36,40)(37,41)(38,63)(39,48)(53,61)(54,60)(55,64)(56,62);(1,4,20,55)(2,9,11,38)(3,14,16,49)(5,25,7,22)(6,21,26,56)(8,29,40,63)(10,35,36,53)(12,32,39,18)(13,33,30,61)(15,44,46,64)(17,51,52,45)(19,27,23,28)(24,54,37,34)(31,58,57,42)(41,59,48,60)(43,50,62,47)Polycyclic presentation:
2^10|a^2=e,b^2=i,d^2=efgj,e^2=i,g^2=j,h^2=j,c^a=cf,d^a=dg,d^b=dfhj,d^c=dfj,e^c=ej,e^d=eh,f^a=fj,f^d=fj,g^a=gh,g^b=gi,g^d=ghj,g^e=gij,h^a=hi,h^d=hi,h^g=hj,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72306331 |
| maximal normal subgroup of | #72306331 |
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.5);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);