Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #20609 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #7742598 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
5595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(3,7)(5,8)(9,10,14,16)(11,20,24,30)(12,17,26,32)(13,19,21,18)(15,28,31,22)(23,37,39,35)(25,34,27,33)(29,36,40,38);(1,2)(3,8,7,5)(4,6)(9,11,21,28)(10,15,18,20)(12,23,25,38)(13,22,14,24)(16,31,19,30)(17,29,34,35)(26,39,27,36)(32,40,33,37);(1,3)(2,5)(4,7)(6,8)(10,16)(11,22)(15,20)(17,32)(18,19)(23,38)(24,28)(29,37)(30,31)(33,34)(35,40)(36,39);(9,12)(10,17)(11,23)(13,27)(14,26)(15,29)(16,32)(18,34)(19,33)(20,35)(21,25)(22,36)(24,39)(28,38)(30,37)(31,40)Polycyclic presentation:
2^10|a^2=j,b^2=eg,e^2=j,b^a=be,c^a=chi,c^b=cf,d^a=dij,d^c=dij,e^a=ej,e^b=ei,e^c=eij,e^d=ej,f^a=fij,f^b=fh,f^d=fj,f^e=fj,g^a=gi,g^b=gi,g^c=ghij,g^d=gj,g^e=gj,g^f=gj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72341467, #72341469, #72341503, … |
| maximal normal subgroup of | #72341467, #72341469, #72341503, … |
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.10);;
gap> SetPower(coll, 2, F.5*F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);