Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#442 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| prime radical | C2×#2184 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C2×#1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,15)(18,22)(23,28)(24,29)(30,31);(1,3,4,7)(2,5,6,8)(9,10,12,15,13,14,16,11)(18,23,22,28)(19,25)(20,27)(24,30,29,31)(26,32);(3,7)(5,8)(9,11,16,14,13,15,12,10)(18,24,22,29)(19,26)(23,30,28,31)(25,32);(2,6)(5,8)(17,18,19,23,20,24,25,31,21,22,26,28,27,29,32,30)Polycyclic presentation:
2^10|b^2=e,c^2=egh,d^2=f,e^2=h,f^2=i,i^2=j,b^a=bh,c^a=ch,c^b=cg,d^a=dgj,d^b=df,d^c=di,e^d=ej,f^b=fi,f^c=fj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72320605, C2×#72320463, #72343708, … |
| maximal normal subgroup of | #72320605, C2×#72320463, #72343708, … |
| maximal quotient of | #72321529, C2×#72320463 |
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.5*F.7*F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);