Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23848 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20385 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(13,27)(15,16)(24,30);(1,2)(3,4)(6,13)(7,17)(8,23)(9,11)(12,27)(15,24)(16,30)(18,29)(19,21)(20,34)(22,31)(25,26)(32,33)(35,36);(5,6,10,12)(7,18,22,35)(8,24,26,30)(9,16,11,15)(13,25,27,23)(14,31,28,17)(19,33,34,32)(20,36,21,29);(2,4)(5,7,8,19,9,20,25,31,10,22,26,34,11,21,23,17)(6,14,13,29,15,32,30,35,12,28,27,36,16,33,24,18)Polycyclic presentation:
2^10|c^2=j,d^2=e,e^2=ij,f^2=j,i^2=j,c^a=ch,c^b=ce,d^a=dg,d^b=defj,d^c=dei,e^b=ei,e^c=ei,f^b=fj,f^c=fj,g^c=gj,h^d=hj,i^b=ij,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250334, #72250398, #72252652, … |
| maximal normal subgroup of | #72250334, #72250398, #72252652, … |
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, 3, F.10);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);