Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12695 |
| commutator | C22×C42 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| prime radical | C2×#3151 | C22 |
| LCS | G ▷ C22×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #34637 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1539 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 1539 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(10,15)(11,19)(12,24)(13,26)(16,30)(17,33)(20,32)(22,23)(25,27)(28,34)(35,37)(39,40);(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,17)(13,18)(14,15)(16,23)(19,32)(21,31)(22,30)(24,34)(25,28)(26,29)(27,33)(35,39)(36,38)(37,40);(1,3)(2,5,6,7)(4,8)(9,11,12,21,14,23,25,38)(10,16,17,31,15,20,28,36)(13,22,27,37,26,19,24,35)(18,32,33,40,29,30,34,39)Polycyclic presentation:
2^10|b^2=hi,c^2=d,d^2=j,e^2=i,f^2=i,g^2=j,b^a=bij,c^a=ce,c^b=cefi,d^a=dg,d^b=dh,e^a=ei,e^b=ej,e^c=egi,e^d=ei,f^a=fi,f^b=fj,f^c=fghi,f^d=fij,g^a=gj,g^c=gi,g^d=gj,h^c=hj,i^c=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72293082 |
| maximal normal subgroup of | #72293082 |
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.8*F.9);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, 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);