Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #11087 |
| commutator | C22×D4 | C2×C42 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #123590 |
| prime radical | C2×#3449 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C22 ▷ C2 ▷ C1 | |
4953 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)(3,13)(4,18,20,9)(5,11)(6,12)(7,10)(8,16)(14,41,43,61)(15,37)(17,48)(19,53,45,62)(21,51,55,33)(22,52,56,34)(23,31)(24,38)(25,35)(26,42)(27,40)(28,36)(29,44,60,46)(30,49)(32,57,50,54)(39,47)(58,64,59,63);(1,3)(2,8)(4,19,20,45)(5,23,25,16)(6,26,28,17)(7,15)(9,32,18,50)(10,31)(11,37,35,13)(12,39,36,30)(14,22)(21,57)(24,47)(27,49)(29,52)(33,62)(34,60)(38,42)(40,48)(41,64,61,63)(43,56)(44,59,46,58)(51,53)(54,55);(1,4)(2,9)(3,14)(5,21)(6,22)(7,20)(8,29)(10,18)(11,33)(12,34)(13,41)(15,43)(16,46)(17,45)(19,26)(23,44)(24,58)(25,55)(27,59)(28,56)(30,32)(31,60)(35,51)(36,52)(37,61)(38,63)(39,50)(40,64)(42,53)(47,57)(48,62)(49,54)Polycyclic presentation:
2^10|a^2=d,b^2=h,b^a=bdf,c^a=ce,c^b=ceg,d^c=dj,e^a=ej,e^b=ej,f^b=fi,f^c=fi,g^a=gi,g^b=gi,g^f=gj,h^a=hi,h^c=hij,h^f=hj,h^g=hj,i^b=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72303733, #72312767, #72313024, … |
| maximal normal subgroup of | #72303733, #72312767, #72313024, … |
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.4);;
gap> SetPower(coll, 2, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7);;
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, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*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);