Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #147779 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| prime radical | #7503371 | C2 |
| 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 | |
2431 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)(4,18,22,48)(5,25)(6,28)(7,11)(9,21,35,52)(10,20)(12,29)(13,41,43,47)(14,31)(15,40)(16,32)(17,33)(19,50,44,63)(23,49,24,56)(26,38)(27,39)(30,42,60,64)(34,59,61,51)(36,54,37,55)(45,57)(46,62)(53,58);(1,3)(2,8)(4,19,23,13)(5,26,20,45)(6,15,29,16)(7,17)(9,34,36,60)(10,38,25,57)(11,33)(12,40,28,32)(14,27)(18,47,49,63)(21,42,54,51)(22,44,24,43)(30,35,61,37)(31,39)(41,56,50,48)(46,53)(52,64,55,59)(58,62);(1,4,20,52)(2,9,10,18)(3,13,26,59)(5,21,7,22)(6,24,53,55)(8,30,57,50)(11,35,25,48)(12,37,39,49)(14,42,15,19)(16,44,46,64)(17,43,45,51)(23,27,54,29)(28,36,58,56)(31,41,32,34)(33,60,38,63)(40,61,62,47)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=dej,d^2=ij,e^2=ij,h^2=j,b^a=bfg,c^a=cd,c^b=ce,d^a=di,d^b=dh,e^a=ej,e^b=ehi,f^c=fj,g^c=gh,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 | #72286575, #72286823, #72287007 |
| maximal normal subgroup of | #72286575, #72286823, #72287007 |
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.10);;
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.6*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);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
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);