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 | #511811 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,8,13,22)(4,15,16,9)(6,11)(7,18)(12,25,26,21)(14,23)(17,20)(19,28)(24,30)(27,32)(29,31);(2,7,10,11)(8,19,22,23)(9,20,15,24)(21,32,25,31);(1,3)(2,8)(4,12)(5,13)(6,14)(7,19)(9,21)(10,22)(11,23)(15,25)(16,26)(17,27)(18,28)(20,32)(24,31)(29,30);(1,4,5,16)(2,9,10,15)(3,12,13,26)(6,17,18,30)(7,20,11,24)(8,21,22,25)(14,29,28,27)(19,31,23,32)Polycyclic presentation:
2^10|a^2=e,b^2=i,d^2=j,g^2=j,b^a=bg,d^a=dj,d^b=df,d^c=dej,e^b=ei,f^a=fh,f^c=fi,g^a=gij,g^d=gh,g^e=gj,h^c=hj,i^a=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72256070, #72262472, #72262507, … |
| maximal normal subgroup of | #72256070, #72262472, #72262507, … |
| maximal quotient of | #72337658, #72337671, #72337704, … |
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.5);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
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, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);