Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12759 |
| commutator | #390 | C22×C4 |
| Frattini, cosocle | C2×#390 | C23 |
| prime radical | #6437608 | C2 |
| LCS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ C2×#390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
3473 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 3473 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,14,19)(11,15,21,31)(12,16,25,34)(13,17,28,18)(20,29,23,39)(22,36,26,30)(24,33,27,35)(32,37,40,38);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,23,25,20)(13,26)(14,21)(16,32,34,40)(17,36)(18,30)(19,31)(22,28)(24,38,27,37)(29,33,39,35);(2,5)(3,8)(6,7)(9,12,13,27,14,25,28,24)(10,16,17,35,19,34,18,33)(11,20,22,38,21,23,26,37)(15,29,36,32,31,39,30,40)Polycyclic presentation:
2^10|a^2=j,b^2=f,c^2=gij,d^2=hj,g^2=j,b^a=be,c^b=cd,d^a=dg,d^b=dh,d^c=dhij,e^c=egij,e^d=eij,f^c=fj,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^e=gj,i^a=ij,i^c=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72338282, #72339625 |
| maximal normal subgroup of | #72338282, #72339625 |
| maximal quotient of | #72338409, #72339794, #72339804 |
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.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
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.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*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, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);