Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #137419 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | C2×#379 | C23 |
| prime radical | #37436 | C22 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1396 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,16,26)(4,17,20,24)(5,10)(6,11)(7,12)(8,15,23,21)(9,18,25,19)(14,22)(27,29)(28,31)(30,32);(1,3)(2,8,12,23)(4,14)(5,21)(6,15)(7,16)(9,22,25,32)(10,26,11,13)(17,31,24,27)(18,29)(19,28)(20,30);(1,4,7,20)(2,9,12,25)(3,14,16,30)(5,18)(6,19)(8,22,23,32)(10,17)(11,24)(13,27)(15,28)(21,29)(26,31)Polycyclic presentation:
2^10|a^2=i,b^2=f,c^2=gj,d^2=hj,e^2=hij,b^a=be,c^a=cd,d^a=dh,d^b=dg,d^c=dhij,e^a=ehi,e^b=ehi,e^c=egi,e^d=ei,f^a=fj,g^a=gi,g^d=gj,g^e=gj,h^b=hj,i^b=ij,i^c=ij58 conjugacy classes — view the full character table.
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.9);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, 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, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);