Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8546 |
| commutator | C24×C4 | C22×C4 |
| Frattini, cosocle | C22×#123 | C23 |
| prime radical | #4435227 | C2 |
| LCS | G ▷ C24×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
3087 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 3087 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,15,7)(4,17,19,8)(5,11)(6,12)(9,14,23,28)(10,20)(16,27,31,22)(18,25,32,26)(21,29,24,30);(2,7)(3,14)(4,18)(8,24)(9,11)(10,23)(12,13)(15,28)(16,31)(17,27)(19,32)(21,25)(22,26)(29,30);(2,8)(3,14)(7,21)(9,22)(10,25)(11,17)(12,26)(13,27)(15,28)(16,29)(23,24)(30,31)Polycyclic presentation:
2^10|a^2=e,d^2=j,b^a=bd,c^a=cf,c^b=cghj,d^a=dg,d^b=dj,d^c=dj,e^b=egj,e^c=eh,e^d=ei,f^a=fh,f^b=fi,f^e=fj,g^a=gi,h^a=hj52 conjugacy classes — view the full character table.
| maximal subgroup of | #72238371, #72238386, #72262532, … |
| maximal normal subgroup of | #72238371, #72238386, #72262532, … |
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, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);