Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #15986 |
| commutator, Frattini, cosocle | C26 | C24 |
| prime radical | #62817 | C22 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
3089 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 3089 conjugacy classes of subgroups — too many to draw. Only its 197 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)(3,7)(4,9)(5,10)(6,11)(8,18)(12,19)(13,21)(14,20)(15,25)(16,23)(17,26)(22,27)(24,28)(29,31)(30,32);(2,7,10,19)(3,12)(4,15)(8,22,11,21)(9,24,25,20)(13,27)(16,17)(23,31,26,32);(2,8)(3,13)(4,16)(7,19)(10,11)(12,27)(14,28)(15,17)(20,31)(21,22)(24,32)(29,30);(2,9,11,26)(4,17)(7,20,21,31)(8,23,10,25)(14,29)(15,16)(19,24,22,32)(28,30)Polycyclic presentation:
2^10|b^2=g,d^2=h,b^a=be,c^a=cij,c^b=ch,d^a=df,d^b=dg,d^c=dgh,e^b=ei,e^c=ej,f^b=fi,f^c=fij,f^d=fj,g^a=gi,h^a=hj70 conjugacy classes — view the full character table.
| maximal subgroup of | #72303408, #72303475, #72305938 |
| maximal normal subgroup of | #72303408, #72303475, #72305938 |
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.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, 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);