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 | #498833 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2710 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)(7,16,20,19)(9,23)(11,18,26,27)(13,24)(17,30,33,31)(22,35)(25,32,36,34)(28,29);(7,17,20,33)(8,21)(9,22)(11,25,26,36)(12,14)(13,29)(16,30,19,31)(18,32,27,34)(23,35)(24,28);(2,4)(5,6,10,15)(7,18,20,27)(8,14,21,12)(9,24,23,13)(11,16,26,19)(17,32,33,34)(22,28,35,29)(25,30,36,31);(2,4)(5,7,9,17,8,16,22,31,10,20,23,33,21,19,35,30)(6,11,24,36,14,18,28,32,15,26,13,25,12,27,29,34)Polycyclic presentation:
2^10|a^2=f,b^2=f,c^2=j,d^2=efgij,e^2=j,g^2=i,i^2=j,c^a=ch,d^a=dhi,d^b=dg,d^c=de,e^b=ej,e^c=ej,e^d=ej,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72249919, #72250233, #72250334, … |
| maximal normal subgroup of | #72249919, #72250233, #72250334, … |
| maximal quotient of | #72265708, C2×#72249884 |
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.6);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
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.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
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.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);