Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #367 |
| commutator | C16 | C4×C16 |
| Frattini, cosocle | C2×C8×C16 | C22 |
| socle | C22 | #7171 |
| prime radical | C23 | #1331 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ C2×C8×C16 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(18,26,28,41)(19,33)(20,34)(21,39)(22,40)(23,24)(27,46,43,29)(30,47,48,42)(31,45,32,44)(35,38)(36,37);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,19,27,21,29,20,26,23,31,34,42,22,30,33,44,25,28,36,43,40,46,38,41,24,32,35,47,39,48,37,45)Polycyclic presentation:
2^10|a^2=e,b^2=c,c^2=f,d^2=g,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bd,c^a=cg,d^a=dgi,e^b=ej,f^a=fi,g^a=gij,h^a=hj,i^a=ij208 conjugacy classes — view the full character table.
| maximal subgroup of | #72227027, #72227131, #72227140 |
| maximal normal subgroup of | #72227027, #72227131, #72227140 |
| maximal quotient of | #72227006 |
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.5);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
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.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, 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, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);