Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #149761 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | #35346 | C4 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,7)(8,21)(9,22)(10,14)(12,18)(23,31)(24,27)(25,32)(26,29)(28,30);(1,2,5,9)(3,8,12,23,13,24,18,25)(4,14,16,29,17,7,19,22)(6,10,20,26)(11,27,28,32)(15,21,30,31);(1,3)(2,8)(4,15,17,11)(5,18,20,12)(6,13)(7,21,14,27)(9,25,26,23)(10,24)(16,28)(19,30)(22,32)(29,31)Polycyclic presentation:
2^10|b^2=e,c^2=ghi,e^2=g,b^a=bd,c^a=cfhi,c^b=cfghij,d^b=df,d^c=dh,e^a=ef,e^c=ehij,e^d=eh,f^b=fh,f^c=fij,f^e=fi,g^a=gi,g^c=gj,g^d=gj,h^b=hi,h^c=hj,h^e=hj,i^b=ij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72283898, #72283924, #72283962, … |
| maximal normal subgroup of | #72283898, #72283924, #72283962, … |
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.5);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, 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);