Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12123 |
| commutator, Frattini, cosocle | C22⋊#95 | C23 |
| prime radical | #103197 | C2 |
| LCS | G ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C22⋊#95 ◁ G | |
| LECS | G ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1197 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(11,17)(14,21)(15,24)(18,19)(22,23);(1,3)(2,6)(4,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,23)(19,22)(20,24);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,14)(12,19,20,17)(15,22)(16,23)(21,24)Polycyclic presentation:
2^10|c^2=ghj,d^2=g,e^2=ghij,b^a=bd,c^a=cdeghij,c^b=cgij,d^a=dg,d^b=dg,d^c=df,e^a=egh,e^b=efgij,e^c=efh,e^d=eij,f^a=fi,f^e=fj,h^b=hj,i^c=ij49 conjugacy classes — view the full character table.
| maximal quotient of | #72267761 |
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, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);