Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1317 |
| commutator | C25 | C22×C8 |
| Frattini, cosocle | C2×#379 | C23 |
| prime radical | C27 | C2×C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C26 ◁ C22×#379 ◁ G | |
| LECS | G ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
5252 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 5252 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11)(7,13,15,19,14,10,16,18)(9,12,17,20);(2,4);(1,2,3,4)(5,7)(6,10,12,13)(8,16)(9,14)(11,19,20,18)(15,17)Polycyclic presentation:
2^10|a^2=f,c^2=dij,f^2=h,c^a=ce,c^b=cd,e^a=eg,e^c=ej,f^c=fg,f^e=fi,g^a=gi,g^f=gj,h^c=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72291978, #72292064, #72292068 |
| maximal normal subgroup of | #72291978, #72292064, #72292068 |
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.6);;
gap> SetPower(coll, 3, F.4*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);