Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1333 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | #33733 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 99 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 — 3 generators on 64 points:
(1,2)(3,8)(4,9)(5,10)(6,7)(11,17)(12,20)(13,21)(14,22)(15,18)(16,19)(23,27)(24,28)(25,30)(26,29)(31,32)(34,41)(35,48)(36,54)(37,49)(38,39)(42,56)(43,46)(44,58)(45,57)(47,59)(50,51)(52,63)(53,62)(55,61)(60,64);(2,7)(3,11)(9,18)(10,19)(12,23)(13,24)(22,29)(25,31)(34,42)(41,56)(43,59)(44,60)(45,55)(46,47)(57,61)(58,64);(1,3,4,12,5,13,14,25,6,11,15,23,16,24,26,31)(2,8,9,20,10,21,22,30,7,17,18,27,19,28,29,32)(33,34,35,43,36,44,50,58,38,46,51,41,37,55,48,57,40,42,53,59,49,60,63,64,39,47,52,56,54,45,62,61)Polycyclic presentation:
2^10|c^2=d,d^2=f,e^2=gij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhi,c^a=ce,c^b=chij,d^a=dg,e^a=eg,e^c=ei,e^d=ej,f^a=fi,g^a=gij,g^c=gj,h^a=hj,i^a=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72287210, #72287236, #72287413, … |
| maximal normal subgroup of | #72287210, #72287236, #72287413, … |
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, 3, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);