Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1781 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | C2×#35527 | C2 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×#127 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
14747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22);(1,2,3,6)(4,5,7,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24);(11,18)(14,23)(17,20)(22,24)Polycyclic presentation:
2^10|b^2=e,b^a=bf,c^a=chi,c^b=cg,d^b=di,e^a=eh,e^c=ei,f^b=fh,f^c=fij,f^e=fj,g^a=gi,g^b=gi,h^b=hj88 conjugacy classes — view the full character table.
| maximal subgroup of | #72226161, #72262276, #72262284, … |
| maximal normal subgroup of | #72226161, #72262276, #72262284, … |
| maximal quotient of | #72226082, #72249475 |
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> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*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, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);