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 | #149741 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C22×#560 | C4 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C26 ◁ #21746 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1612 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1612 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,7)(4,15)(5,10)(6,8)(9,16)(11,29)(12,22)(13,23)(14,24)(17,28)(18,27)(19,26)(20,25)(21,31)(30,32);(1,3,4,12,6,14,16,30)(2,7,9,22,8,24,15,32)(5,18,19,31)(10,27,28,21)(11,20,13,17)(23,26,29,25);(2,8)(3,11)(4,16)(7,21)(10,25)(12,18)(13,30)(14,31)(17,19)(22,23)(24,29)(27,32)Polycyclic presentation:
2^10|b^2=d,d^2=f,e^2=fh,b^a=bfhij,c^a=cj,c^b=cdefh,d^a=dij,d^c=dg,e^b=egij,e^c=egj,e^d=ehj,f^c=fi,f^e=fj,g^b=gh,g^d=gi,g^e=gi,h^b=hi,h^d=hj,h^e=hj,i^b=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72340890, #72341180 |
| maximal normal subgroup of | #72340890, #72341180 |
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.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);