Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #21452 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20436 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2586 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)(7,15,17,32)(8,19)(9,22)(11,23,26,35)(13,29)(14,28)(16,31,33,18)(20,21)(24,34,36,27)(25,30);(7,16)(8,20)(9,22)(11,24)(13,30)(14,28)(15,31)(17,33)(18,32)(19,21)(23,34)(25,29)(26,36)(27,35);(2,4)(5,6,10,12)(7,11)(8,13,21,25)(9,14,22,28)(15,23)(16,24)(17,26)(18,27)(19,29,20,30)(31,34)(32,35)(33,36);(2,4)(5,7,8,15,9,18,19,33,10,17,21,32,22,31,20,16)(6,11,25,35,14,27,30,24,12,26,13,23,28,34,29,36)Polycyclic presentation:
2^10|a^2=f,c^2=fj,d^2=efgh,g^2=i,i^2=j,c^a=ch,d^a=dg,d^b=dghij,d^c=de,e^c=ej,e^d=ej,f^d=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72249919, #72250235, #72250330, … |
| maximal normal subgroup of | #72249919, #72250235, #72250330, … |
| maximal quotient of | #72265685, C2×#72249887 |
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.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*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);