Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #379754 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #6437621 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
5349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,20)(6,10)(7,11)(8,16)(13,28)(14,23)(15,22)(17,24)(18,25)(19,26)(21,31)(27,30)(29,32);(1,3,6,16,7,15,5,14)(2,8,10,22,11,23,20,12)(4,13,18,29,19,30,17,31)(9,21,25,28,26,32,24,27);(1,4)(2,9)(3,13,15,30)(5,17)(6,18)(7,19)(8,21,23,32)(10,24)(11,26)(12,27,22,28)(14,29,16,31)(20,25);(1,5,7,6)(2,10,11,20)(3,14,15,16)(4,17,19,18)(8,12,23,22)(9,24,26,25)(13,29,30,31)(21,27,32,28)Polycyclic presentation:
2^10|b^2=g,c^2=fgh,d^2=j,f^2=j,g^2=j,b^a=bf,c^a=ce,c^b=ceh,d^a=dhij,d^b=dh,e^b=ei,e^d=ej,f^a=fj,f^b=fj,f^c=fij,f^e=fj,g^c=gi,g^e=gj,h^a=hj,h^c=hj,i^a=ij,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72256116, #72256121, #72256177, … |
| maximal normal subgroup of | #72256116, #72256121, #72256177, … |
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.7);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*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);