Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24196 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#509 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20438 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12,16,15)(7,17,22,23)(8,24)(13,26,30,28)(14,31)(18,27)(19,34,21,32)(20,25);(1,2)(3,4)(5,6,8,13,10,12,20,28,11,16,24,30,9,15,25,26)(7,18,21,35,23,14,32,33,22,27,19,36,17,31,34,29);(6,13,16,30)(7,19,22,21)(8,20)(9,10)(12,26,15,28)(14,27)(17,34,23,32)(18,31)(24,25)(29,33);(2,4)(5,7,20,34,9,17,8,19,11,22,25,32,10,23,24,21)(6,14,26,36,15,18,30,33,16,31,28,35,12,27,13,29)Polycyclic presentation:
2^10|a^2=g,b^2=e,c^2=g,d^2=egi,e^2=i,f^2=j,h^2=j,i^2=j,b^a=bh,c^b=ceij,d^a=dghj,d^b=df,d^c=dej,e^a=ej,e^c=ei,e^d=ej,f^c=fj,g^b=gj,g^d=gj,h^c=hj,h^d=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250777, #72250983, #72251004, … |
| maximal normal subgroup of | #72250777, #72250983, #72251004, … |
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> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*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, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);