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 | #6437451 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24×C4 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3632 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)(4,18,22,56,19,36,24,40)(5,25,21,41)(6,12)(7,10)(9,23,35,26,38,57,37,58)(11,28,34,27)(14,45,47,62,46,64,20,55)(16,51)(17,44,50,32)(29,39)(30,48,61,59,42,63,33,52)(31,54)(43,53)(49,60);(4,19)(5,21)(11,34)(14,46)(16,49)(18,36)(20,47)(22,24)(25,41)(27,28)(31,43)(40,56)(45,64)(51,60)(53,54)(55,62);(1,3)(2,8)(4,20,19,47)(5,16)(6,17)(7,15)(9,33,38,61)(10,13)(11,31)(12,32)(14,22,46,24)(18,55,36,62)(21,49)(23,59,57,52)(25,60)(26,48,58,63)(27,53)(28,54)(29,50)(30,35,42,37)(34,43)(39,44)(40,45,56,64)(41,51);(1,4,21,58)(2,9,34,56)(3,14,16,52)(5,26,7,19)(6,22,27,23)(8,30,31,55)(10,38,11,40)(12,35,41,36)(13,42,43,62)(15,46,49,59)(17,20,54,63)(18,39,37,25)(24,28,57,29)(32,33,60,64)(44,61,51,45)(47,53,48,50)Polycyclic presentation:
2^10|a^2=e,c^2=i,d^2=egj,e^2=i,g^2=j,h^2=j,c^a=cf,d^a=dfgj,d^b=df,d^c=dhj,e^c=ej,e^d=eh,f^a=fj,f^d=fj,g^a=ghj,g^b=gj,g^c=gi,g^d=gh,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, #72306394, #72318686 |
| maximal normal subgroup of | #72306331, #72306394, #72318686 |
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, 3, F.9);;
gap> SetPower(coll, 4, F.5*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.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*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*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
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);