Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #149280 |
| commutator | #532 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#532 | C23 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1143 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)(4,17,21,43)(5,23)(6,26)(7,11)(9,20,35,50)(10,19)(12,28)(13,39,41,64)(14,30)(15,32)(16,33)(18,46,48,63)(22,44,55,45)(24,37)(25,38)(27,31)(29,40,59,52)(34,62,60,47)(36,53,58,54)(42,61)(49,56)(51,57);(1,3)(2,8)(4,18,22,13)(5,24,19,49)(6,27,28,15)(7,16)(9,34,36,59)(10,37,23,56)(11,33)(12,31,26,32)(14,25)(17,39,44,46)(20,52,53,62)(21,48,55,41)(29,35,60,58)(30,38)(40,54,47,50)(42,51)(43,64,45,63)(57,61);(1,4,19,50)(2,9,10,17)(3,13,24,47)(5,20,7,21)(6,22,51,53)(8,29,56,63)(11,35,23,43)(12,36,38,45)(14,40,27,48)(15,18,42,52)(16,41,49,62)(25,54,28,55)(26,58,57,44)(30,39,32,60)(31,34,61,64)(33,59,37,46)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=dei,d^2=ij,e^2=ij,h^2=j,b^a=bg,c^a=cd,c^b=ce,d^a=di,d^b=dh,e^a=ej,e^b=ehij,f^c=fj,g^c=ghj,g^d=gi,g^e=gij,h^a=hi,h^b=hij,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286685, #72286823, #72287086 |
| maximal normal subgroup of | #72286685, #72286823, #72287086 |
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.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.4*F.5*F.9);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);