Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #21746 |
| commutator | C23×C4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #4418479 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×D42 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
4840 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 4840 conjugacy classes of subgroups — too many to draw. Only its 221 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8,15,24)(4,9,17,7)(5,11)(6,10)(12,25,16,26)(13,22)(14,23,19,28)(18,27,29,20)(21,31,32,30);(2,7)(3,12)(4,17)(8,21)(9,11)(10,20)(14,30)(15,16)(18,29)(19,31)(22,27)(23,24)(25,32)(26,28);(4,18)(7,20)(9,27)(14,19)(17,29)(21,32)(23,28)(30,31);(1,3,13,15)(2,8,22,24)(4,19,29,31)(5,16,6,12)(7,21,27,23)(9,28,20,32)(10,25,11,26)(14,17,30,18)Polycyclic presentation:
2^10|a^2=f,d^2=ij,e^2=j,b^a=be,c^b=cg,d^a=dghi,d^b=dgi,e^a=eh,e^b=ej,e^c=ei,e^d=ei,f^b=fhj,f^d=fij,f^e=fj,g^a=gi,h^a=hj64 conjugacy classes — view the full character table.
| maximal subgroup of | #72263157, #72263167, #72263233, … |
| maximal normal subgroup of | #72263157, #72263167, #72263233, … |
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.6);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);