Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D16 |
| commutator | C16 | C2×C32 |
| Frattini, cosocle | C162 | C22 |
| socle | C22 | #7068 |
| prime radical | C2×D16 | C16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ C162 ◁ G | |
| LECS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
145 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 512 |
A minimal faithful permutation representation — 2 generators on 64 points:
(33,34)(35,39)(36,40)(37,42)(38,44)(41,58)(43,61)(45,51)(46,52)(47,53)(48,54)(49,56)(50,57)(55,64)(59,62)(60,63);(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32)(34,39,40,52,42,53,56,59,44,54,57,60,45,55,43,41)Polycyclic presentation:
2^10|b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bd,c^a=cf,d^a=dfhj,e^a=eh,f^a=fhj,g^a=gj,h^a=hj304 conjugacy classes — view the full character table.
| maximal subgroup of | #72226995, #72227075, #72227079 |
| maximal normal subgroup of | #72226995, #72227075, #72227079 |
| maximal quotient of | #72226989, #72226990 |
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, 2, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);