Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24196 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20436 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,10,17,23,36,11,15,30,35,22,7,18,21)(8,19,25,34,24,13,31,20,26,32,9,14,29,33,28,27);(2,4)(5,7,10,15,11,17,22,6)(8,19,24,33,26,32,29,13)(9,27,28,14,25,20,31,34)(12,16,23,21,30,35,18,36);(1,2,3,4)(5,8,22,29,11,26,10,24)(6,13,17,32,15,33,7,19)(9,18,31,30,25,23,28,12)(14,16,27,36,34,35,20,21);(5,9)(6,14,15,34)(7,20,17,27)(8,23)(10,28)(11,25)(12,29)(13,16,33,35)(18,26)(19,21,32,36)(22,31)(24,30)Polycyclic presentation:
2^10|a^2=h,b^2=e,c^2=efj,d^2=eij,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cij,c^b=cf,d^a=dfg,d^b=dfgj,d^c=dei,e^a=ej,g^c=gj,g^d=gj,h^b=hi,h^c=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72320154, #72322329, #72322804, … |
| maximal normal subgroup of | #72320154, #72322329, #72322804, … |
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.5);;
gap> SetPower(coll, 3, F.5*F.6*F.10);;
gap> SetPower(coll, 4, F.5*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);