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 | #24338 |
| 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 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(7,16,20,35)(8,19)(9,25)(10,24)(12,27,30,36)(13,33)(14,34)(17,23,22,21)(18,32,28,31)(26,29);(7,17,20,22)(8,9)(12,28,30,18)(13,26)(16,23,35,21)(19,25)(27,31,36,32)(29,33);(1,2,4,3)(5,6)(7,18,20,28)(8,13)(9,26)(10,14)(11,15)(12,17,30,22)(16,31,35,32)(19,33)(21,36,23,27)(24,34)(25,29);(1,3,4,2)(5,7,19,35,24,17,8,23,11,20,25,16,10,22,9,21)(6,12,29,27,34,18,26,31,15,30,33,36,14,28,13,32)Polycyclic presentation:
2^10|a^2=g,b^2=g,c^2=e,d^2=eghij,f^2=i,h^2=i,i^2=j,c^a=ce,c^b=cg,d^a=defij,d^b=dfgh,d^c=dij,e^d=ej,f^a=fi,f^b=fj,f^d=fj,g^d=gj,h^a=hi,h^b=hj,h^c=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72249855, #72252637, #72252718, … |
| maximal normal subgroup of | #72249855, #72252637, #72252718, … |
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.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*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);