Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#535 |
| commutator | C22×C4 | C24×C4 |
| Frattini, cosocle | C23×C4 | C25 |
| socle | C22 | #34963 |
| prime radical | C2×#3067 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,17,12,19,31,18)(7,13,22,10,23,33,16,34)(8,24,27,15,28,14,9,29)(20,36,30,35,25,32,21,26);(5,7)(6,13)(8,25)(9,30)(10,17)(11,22)(12,23)(14,36)(15,26)(16,31)(18,34)(19,33)(20,28)(21,27)(24,32)(29,35);(1,2,3,4)(5,8,11,27,12,28,31,9)(6,14,17,29,19,24,18,15)(7,20,22,30,23,25,16,21)(10,26,33,36,34,35,13,32);(5,9,31,28,12,27,11,8)(6,15,17,14,19,29,18,24)(7,21,22,20,23,30,16,25)(10,32,13,35,34,36,33,26);(5,10,11,33,12,34,31,13)(6,16,18,23,19,22,17,7)(8,26,27,36,28,35,9,32)(14,21,15,25,24,30,29,20)Polycyclic presentation:
2^10|a^2=f,c^2=fg,d^2=ij,e^2=i,f^2=j,i^2=j,c^a=cj,c^b=cj,d^a=dh,d^b=dh,d^c=dfij,e^a=ei,e^b=eij,h^c=hj,h^d=hj,i^a=ij,i^b=ij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72321642, #72322146, #72322466, … |
| maximal normal subgroup of | #72321642, #72322146, #72322466, … |
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, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);