Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×C22⋊C4 |
| commutator | C23 | C25×C4 |
| Frattini, cosocle | C24 | C26 |
| prime radical | C2×#62817 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×#120 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
35913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2)(3,5)(4,6)(7,8);(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(1,3)(2,5)(4,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,6)(5,8)(11,18)(14,22)(17,24)(21,23);(3,7)(5,8)Polycyclic presentation:
2^10|a^2=g,d^a=dh,e^b=ei,e^c=ej,f^d=fi,g^d=gj,h^a=hj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72234599, #72234665, #72237564, … |
| maximal normal subgroup of | #72234599, #72234665, #72237564, … |
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.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);