Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1629 |
| commutator | C4×C8 | C22×C8 |
| Frattini, cosocle | C42×C8 | C23 |
| socle | C2 | #99341 |
| prime radical | #3147 | C2×C4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,14,8,13,31,34,9,15,16,12,7,33,5,30)(3,10,20,36,19,37,47,44,22,39,53,60,21,38,18,45)(4,23,26,43,28,46,25,58,29,56,32,11,27,41,59,42)(17,48,50,62,49,64,24,55,52,61,54,63,51,40,57,35);(1,3,16,47,8,21,5,20,9,22,6,18,7,19,31,53)(2,10,34,60,33,37,14,45,15,39,30,36,13,38,12,44)(4,24,32,51,28,50,59,52,29,57,26,49,27,54,25,17)(11,40,23,55,58,48,41,63,43,64,56,35,42,61,46,62);(1,4,5,25,7,27,16,26,9,29,31,59,8,28,6,32)(2,11,12,41,13,42,30,23,15,43,14,46,33,58,34,56)(3,17,18,24,21,51,53,50,22,52,47,57,19,49,20,54)(10,35,36,61,37,62,44,40,39,55,60,48,38,63,45,64)Polycyclic presentation:
2^10|a^2=f,b^2=efhij,c^2=d,d^2=g,e^2=hi,f^2=i,g^2=j,h^2=j,i^2=j,b^a=be,c^a=cef,c^b=cfh,d^a=dh,d^b=di,e^a=ehj,e^b=eij,f^b=fij,f^c=fj,g^a=gj,g^b=gj,h^a=hj,i^b=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72225483, #72225498 |
| maximal normal subgroup of | #72225483, #72225498 |
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.5*F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);