Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13346 |
| commutator, Frattini, cosocle | #2285 | C23 |
| prime radical | #234066 | C2 |
| LCS | G ▷ #2285 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #2285 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1402 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(3,7)(5,8)(10,14)(11,17)(12,21)(15,20)(16,22)(19,23);(2,5,6,8)(9,10)(11,18,17,23,20,24,15,19)(12,22)(13,14);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19,22,23)(13,20)(14,17)(16,18,21,24)Polycyclic presentation:
2^10|b^2=ej,c^2=fgh,d^2=hij,e^2=ij,f^2=j,g^2=j,b^a=be,c^a=cf,c^b=cd,d^a=dg,d^b=dgh,d^c=dh,e^a=eij,e^c=egi,e^d=eij,f^a=fj,f^b=fj,f^c=fj,g^a=gj,g^b=gi,g^d=gj,h^c=hj,i^c=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72339737 |
| maximal normal subgroup of | #72339737 |
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.5*F.10);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);