Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13561 |
| commutator | #127 | C2×C42 |
| Frattini, cosocle | C2×#493 | C23 |
| prime radical | C2×#1619 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ C43 ◁ #21690 ◁ G | |
| LECS | G ▷ C2×#493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11,9,12,17,19,10,14,7,15,18,13,16,20);(5,7,9,16,10,8,18,17)(6,11)(12,20)(13,19)(14,15);(1,2,3,4)(6,12)(7,16)(8,17)(9,18)(11,15)(13,14)Polycyclic presentation:
2^10|a^2=e,b^2=gh,c^2=f,e^2=hi,g^2=i,h^2=j,b^a=bd,c^a=cgi,c^b=chj,d^a=dg,d^b=dh,e^b=egi,e^c=ehi,e^d=ehij,g^a=gh,g^b=gi,g^c=gi,g^d=gi,h^a=hj,h^b=hi,h^c=hj,h^d=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72279743, #72280016 |
| maximal normal subgroup of | #72279743, #72280016 |
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.5);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);