Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #110030 |
| commutator | C22×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C24×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
2243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10)(3,13,15,24)(4,18,19,26)(6,12,7,11)(8,16,23,17)(9,21,25,20)(14,28,29,22)(27,31,32,30);(1,3,6,15,7,17,5,16)(2,8,10,23,11,13,12,24)(4,14,20,29,21,31,19,30)(9,22,25,28,26,27,18,32);(1,4)(2,9,11,26)(3,14)(5,19)(6,20)(7,21)(8,22,13,27)(10,25,12,18)(15,29)(16,30)(17,31)(23,28,24,32)Polycyclic presentation:
2^10|a^2=d,b^2=g,c^2=e,f^2=j,g^2=j,b^a=bdfj,c^a=cdh,d^b=di,e^a=ej,f^a=fi,f^b=fh,f^c=fij,f^d=fj,g^a=ghi,g^d=gj,h^c=hj,i^a=ij,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72256138, #72256147, #72256166, … |
| maximal normal subgroup of | #72256138, #72256147, #72256166, … |
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.4);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);