Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23764 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#472 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(7,15,19,33)(8,18)(12,26,24,35)(13,25)(16,30,34,27)(17,31,28,36)(20,32)(21,29);(1,2)(3,4)(6,11)(7,16,19,34)(8,20)(9,22)(12,17,24,28)(13,21)(15,30,33,27)(18,32)(25,29)(26,31,35,36);(2,4)(5,6,9,23,10,11,22,14)(7,17,33,36,19,28,15,31)(8,21,20,25,18,29,32,13)(12,27,26,16,24,30,35,34);(5,7,18,34,9,15,32,30,10,19,8,16,22,33,20,27)(6,12,13,28,14,26,29,31,11,24,25,17,23,35,21,36)Polycyclic presentation:
2^10|a^2=h,b^2=h,c^2=gi,d^2=eghj,e^2=i,i^2=j,c^a=cf,c^b=cfg,d^a=di,d^b=de,d^c=degh,e^a=ej,e^b=ei,e^c=eij,g^c=gj,g^d=gj,h^d=hj,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250361, #72257324, #72257552, … |
| maximal normal subgroup of | #72250361, #72257324, #72257552, … |
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.8);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, 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);