Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #9920 |
| commutator, Frattini, cosocle | C23×M16 | C23 |
| prime radical | #394187 | C2 |
| LCS | G ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
2078 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2078 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(9,10,15,16,14,20,13,19)(11,21,26,35,25,31,24,17)(12,27,28,37,23,18,30,32)(22,38,40,36,29,33,39,34);(2,6)(3,7)(5,8)(10,16)(11,22)(12,28)(13,15)(17,33)(18,27)(19,20)(21,36)(23,30)(24,40)(25,29)(26,39)(31,34)(35,38);(1,2,7,8,4,5,3,6)(9,11,23,22,13,26,28,40,14,25,12,29,15,24,30,39)(10,17,27,34,19,21,32,38,20,35,18,36,16,31,37,33)Polycyclic presentation:
2^10|a^2=ghj,c^2=dg,d^2=hj,e^2=j,h^2=j,i^2=j,b^a=be,c^a=cef,c^b=cd,d^a=dg,d^b=dh,d^c=dj,e^b=ej,e^c=ehij,f^b=fghij,f^c=fghi,f^d=fj,g^c=gj,h^b=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72265358 |
| maximal normal subgroup of | #72265358 |
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.7*F.8*F.10);;
gap> SetPower(coll, 3, F.4*F.7);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*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.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);