Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #11174 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C22×#41 | C24 |
| socle | C2 | C2×#11174 |
| prime radical | C2×#3449 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
5017 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)(4,17)(5,22)(6,26)(7,10)(9,19)(11,25)(12,28)(13,39)(14,36)(15,32)(16,30)(18,43)(20,34)(21,35)(23,31)(24,37)(27,38)(29,40)(33,46)(41,56)(42,62)(44,48)(45,49)(47,58)(50,59)(51,64)(52,55)(53,61)(54,57)(60,63);(1,3)(2,8)(4,18,19,46)(5,23)(6,27,28,15)(7,16)(9,33,17,43)(10,30)(11,36)(12,38,26,32)(13,21)(14,25)(20,50,48,47)(22,31)(24,42,55,54)(29,35)(34,59,44,58)(37,57,52,62)(39,45)(40,49)(41,63)(51,53)(56,64)(60,61);(1,4,5,20,7,19,25,48)(2,9,11,34,10,17,22,44)(3,13,14,41,16,40,23,53)(6,21,24,51,28,49,55,63)(8,29,31,56,30,39,36,61)(12,35,37,60,26,45,52,64)(15,18,42,47,27,46,54,50)(32,33,57,58,38,43,62,59);(2,10)(4,19)(8,30)(11,22)(12,26)(13,40)(18,46)(20,48)(21,49)(31,36)(32,38)(37,52)(41,53)(47,50)(51,63)(57,62)Polycyclic presentation:
2^10|b^2=h,c^2=e,e^2=j,f^2=j,b^a=bg,c^a=cgh,c^b=cf,d^a=dj,d^c=dj,e^a=ej,e^b=ej,f^a=fi,f^b=fij,g^c=gij,g^f=gj,h^c=hi,h^f=hj,i^b=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72303363, #72303460, #72303514 |
| maximal normal subgroup of | #72303363, #72303460, #72303514 |
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, 2, F.8);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);