Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #251827 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| prime radical | #511790 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2567 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)(3,8)(4,9)(5,17)(6,10)(7,11)(12,21)(13,27)(14,23)(15,24)(16,25)(18,26)(19,30)(20,22)(28,32)(29,31);(1,3,7,13,6,5,18,20)(2,8,11,22,10,17,26,27)(4,12,16,28,15,19,29,14)(9,21,25,23,24,30,31,32);(1,4)(2,9)(3,12)(5,19)(6,15)(7,16)(8,21)(10,24)(11,25)(13,28)(14,20)(17,30)(18,29)(22,23)(26,31)(27,32);(1,5)(2,8)(3,6)(4,14)(7,13)(9,23)(10,17)(11,27)(12,16)(15,28)(18,20)(19,29)(21,31)(22,26)(24,32)(25,30)Polycyclic presentation:
2^10|b^2=g,g^2=j,b^a=be,d^a=def,d^b=dfgi,d^c=deg,e^b=ei,e^d=eh,f^b=fh,f^c=fij,f^d=fh,g^a=gi,g^d=gh,g^e=gj,h^c=hj,i^b=ij,i^d=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72256292, #72262472, #72262507, … |
| maximal normal subgroup of | #72256292, #72262472, #72262507, … |
| maximal quotient of | #72337666, #72337672, #72337706, … |
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, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
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, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);