Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#7426 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C24×D4 | D4 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C26 ◁ C2×#1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1483 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1483 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,9,17,22)(5,18,20,10)(6,7)(8,13)(12,24)(14,30,28,27)(15,21)(16,25)(19,26)(23,29,32,31);(2,7)(3,12)(9,22)(10,18)(13,15)(14,28)(25,26)(29,31);(1,3,5,14)(2,8,10,23)(4,15,16,31)(6,12,20,28)(7,21,18,32)(9,24,25,30)(11,26,27,22)(13,19,29,17)Polycyclic presentation:
2^10|a^2=fi,c^2=e,d^2=ghj,b^a=bj,c^a=cd,c^b=cj,d^a=dg,d^c=df,e^a=efghi,e^d=eh,f^c=fh,f^d=fij,f^e=fj,g^a=gij,g^c=gi,g^d=gj,g^e=gj,h^a=hi,h^c=hj,h^d=hj,i^c=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72340320, #72340375, #72340379, … |
| maximal normal subgroup of | #72340320, #72340375, #72340379, … |
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*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
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*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, 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, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);