Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12067 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C2×D16 | C2×C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #7053 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,7,8,11,14,6,10,16,3,4,13,15,12)(18,23,25,35,26,37,27,38)(19,28,30,41)(20,31)(24,34,39,47,40,48,42,36)(29,43,45,44)(32,46);(1,3,11,2,4,14,5,13,6,9,15,10,7,12,16,8)(18,24,27,42,26,40,25,39)(19,29,30,45)(20,32)(21,33)(23,34,38,36,37,48,35,47)(28,43,41,44)(31,46);(1,4,6,7)(2,8,10,13)(3,12,9,14)(5,15,16,11)(17,18,19,24,20,23,29,36,21,27,28,39,32,38,44,34,22,26,30,40,31,37,45,47,33,25,41,42,46,35,43,48)Polycyclic presentation:
2^10|a^2=d,b^2=df,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,c^a=cgh,c^b=ce,d^c=di,e^a=ei,e^b=eg,e^d=ej,f^c=fj,g^a=gj,g^b=gi,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72287159, #72287265, #72287277, … |
| maximal normal subgroup of | #72287159, #72287265, #72287277, … |
| maximal quotient of | #72287051 |
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.4*F.6);;
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, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*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.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);