Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #442 |
| commutator | C8 | C22×C32 |
| Frattini, cosocle | C8×C16 | C23 |
| socle | C22 | #12062 |
| prime radical | C2×D16 | C16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ #7331 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(2,6,8,13)(3,9)(4,11)(7,12,15,14)(10,16)(17,18,20,25,21,26,33,41,22,27,34,32,36,43,48,47,23,28,35,42,37,44,29,39,38,45,46,19,24,30,40,31);(2,7,8,15)(3,10)(6,12,13,14)(9,16)(17,19,29,28,36,41,20,30,38,42,48,27,21,31,46,44,23,32,33,18,24,39,35,43,22,25,40,45,37,47,34,26);(1,2,3,6,4,7,9,14,5,8,10,13,11,15,16,12)Polycyclic presentation:
2^10|a^2=d,b^2=dfj,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,c^a=ce,c^b=cg,d^c=di,e^a=eg,e^b=ei,g^a=gi184 conjugacy classes — view the full character table.
| maximal subgroup of | #72230715 |
| maximal normal subgroup of | #72230715 |
| maximal quotient of | #72230715, #72230758 |
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*F.10);;
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, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);