Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×C42 |
| commutator | C2×C4 | C2×C4×C16 |
| Frattini, cosocle | C8×C16 | C23 |
| socle | C22 | C2×#1312 |
| prime radical | C23 | #1312 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 201 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:
(2,6,7,9)(3,10)(8,16,15,14)(11,13)(17,18,20,25,21,24,32,36,22,26,33,38,35,37,45,46,23,27,34,39,19,28,29,41,30,40,43,47,31,42,44,48);(1,2,3,8,4,9,11,16,5,7,10,15,12,6,13,14);(17,19,22,31,23,21,30,35)(18,24,26,37,27,28,40,42)(20,29,33,44,34,32,43,45)(25,36,38,46,39,41,47,48)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bgi,c^a=cj,d^b=di,e^a=ei256 conjugacy classes — view the full character table.
| maximal subgroup of | #72225139, #72225142 |
| maximal normal subgroup of | #72225139, #72225142 |
| maximal quotient of | #72225142 |
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.5);;
gap> SetPower(coll, 3, F.8);;
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.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);