Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13422 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| socle | C2 | #148237 |
| prime radical | #60639 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1432 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)(4,17,20,9)(5,22)(6,12)(7,11)(10,25)(13,41,43,61)(14,45)(15,31)(16,33)(18,50,46,62)(19,35,24,38)(21,49,56,36)(23,37)(26,58)(27,32)(28,40)(29,42,60,44)(30,47)(34,52,48,51)(39,53)(54,63,55,64)(57,59);(1,3)(2,8)(4,18,21,52)(5,23,25,59)(6,27,28,15)(7,16)(9,34,36,62)(10,37,22,57)(11,33)(12,31,40,32)(13,19,44,54)(14,26)(17,48,49,50)(20,46,56,51)(24,42,55,43)(29,35,41,63)(30,39)(38,61,64,60)(45,58)(47,53);(1,4,6,21,7,20,28,56)(2,9,12,36,11,17,40,49)(3,13,27,44,16,43,15,42)(5,24,26,54,25,19,53,55)(8,29,31,41,33,60,32,61)(10,38,39,63,22,35,58,64)(14,46,59,51,47,18,23,52)(30,48,57,50,45,34,37,62)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=hi,d^2=i,e^2=i,h^2=j,b^a=bg,c^a=ce,c^b=cd,d^a=dj,d^b=dh,d^c=dj,e^a=eij,e^b=ehi,f^c=fj,g^c=ghij,g^d=gij,g^e=gi,h^a=hi,h^b=hi,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72284990, #72286685, #72287010 |
| maximal normal subgroup of | #72284990, #72286685, #72287010 |
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.8*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);