Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #9495 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×#127 | C23 |
| prime radical | #245233 | C2 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
2050 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 2050 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,12,15)(3,9,10,16,11,6,13,14)(17,18)(19,25,26,32)(20,24)(21,22)(23,27,30,28)(29,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(18,22)(24,31)(25,32)(27,28);(6,9)(10,13)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,31)(27,29)Polycyclic presentation:
2^10|a^2=d,d^2=i,f^2=j,b^a=bij,c^a=cf,c^b=ce,d^c=dh,e^a=eg,e^d=ei,f^a=fhj,f^b=fgj,f^c=fj,f^e=fj,g^a=gi,g^c=gj,h^b=hi49 conjugacy classes — view the full character table.
| maximal subgroup of | #72337651, #72337663, #72337666, … |
| maximal normal subgroup of | #72337651, #72337663, #72337666, … |
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.4);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);