Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #134037 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #2308 | C23 |
| prime radical | #6432953 | C2 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #35350 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
2710 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2710 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(2,7)(8,21)(9,24)(10,18)(11,27)(12,15)(22,26)(23,31)(25,32)(28,29);(1,2,4,9)(3,11,12,25,14,8,15,23)(5,18,19,24)(6,10,17,26)(7,16,22,20)(13,27,29,32,30,21,28,31);(1,3)(2,8)(4,15,17,12)(5,13)(6,14)(7,21)(9,25,26,23)(10,11)(16,29,19,28)(18,27)(20,30)(22,32,24,31)Polycyclic presentation:
2^10|b^2=d,c^2=ghi,d^2=gij,b^a=be,c^a=cghi,c^b=cf,d^a=dg,d^c=dh,e^b=eg,e^d=ej,f^a=fij,f^b=fh,f^c=fi,f^d=fi,f^e=fj,g^b=gj,g^c=gj,h^b=hi,h^c=hj,h^d=hj,i^b=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72338698, #72338709, #72338737, … |
| maximal normal subgroup of | #72338698, #72338709, #72338737, … |
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, 2, F.4);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);