Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23764 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C23×D8 | C23 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20436 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1383 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)(5,6,9,14,11,15,10,28)(7,16,21,18,22,34,27,35)(8,23,26,30,19,25,24,13)(12,29,31,20,32,36,33,17);(1,2)(3,4)(7,17,22,20)(8,24)(9,10)(12,16,32,34)(13,23)(14,28)(18,31,35,33)(19,26)(21,36,27,29)(25,30);(2,4)(5,6,9,14,11,15,10,28)(7,18,27,16,22,35,21,34)(8,25,26,13,19,23,24,30)(12,17,33,36,32,20,31,29);(5,7,19,17,10,21,26,29,11,22,8,20,9,27,24,36)(6,12,30,34,14,33,25,18,15,32,13,16,28,31,23,35)Polycyclic presentation:
2^10|a^2=f,b^2=h,c^2=gh,d^2=ej,e^2=i,f^2=j,g^2=j,i^2=j,b^a=bfij,c^a=cfgj,c^b=cfij,d^a=de,d^b=deghij,d^c=deij,e^a=ei,e^b=ei,e^c=eij,f^b=fj,f^d=fj,g^b=gj,g^d=gj,h^d=hj,i^a=ij,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250361, #72257203, #72257513, … |
| maximal normal subgroup of | #72250361, #72257203, #72257513, … |
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.6);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.5*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*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);