Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1333 |
| commutator | C16 | C82 |
| Frattini, cosocle | C4×#346 | C22 |
| socle | C2 | #93270 |
| prime radical | C23 | #1388 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ C4×#346 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,13,6,16,26,15,9,17,7,34,8,14,27,11)(3,18,21,50,22,53,23,59,24,54,19,52,35,51,55,44)(5,28,25,48,31,63,61,46,33,43,30,12,32,45,62,47)(10,36,38,60,41,64,37,20,42,29,40,56,39,57,49,58);(1,3,4,19,6,22,26,55,9,24,7,21,8,35,27,23)(2,10,11,37,14,39,34,38,17,42,15,49,16,41,13,40)(5,29,30,56,31,57,62,58,33,36,25,60,32,64,61,20)(12,44,28,51,46,52,45,54,48,59,43,53,47,50,63,18)Polycyclic presentation:
2^10|a^2=e,b^2=c,c^2=f,d^2=gij,e^2=h,f^2=j,g^2=ij,h^2=j,i^2=j,b^a=bd,c^a=ci,d^a=dg,d^b=dgij,d^c=dj,e^b=ej,f^a=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72225320, #72225330, #72225333, … |
| maximal normal subgroup of | #72225320, #72225330, #72225333, … |
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.5);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);