Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | C2×#3415 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17316 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,16)(12,24)(13,17)(14,18)(15,19)(20,31)(21,32)(22,34)(23,35)(25,39)(26,33)(27,40)(28,38)(29,36)(30,37);(1,2,4,3,5,6,8,7)(11,20,12,25)(13,29)(14,26)(16,31,24,39)(17,36)(18,33)(21,30)(22,23,27,28)(32,37)(34,35,40,38);(2,6)(3,7)(9,11,21,25,14,23,30,27,15,12,13,20,26,28,29,22)(10,16,32,39,18,35,37,40,19,24,17,31,33,38,36,34);(1,3,4,6,5,7,8,2)(9,12,26,28,15,11,14,23)(10,16,33,35,19,24,18,38)(13,27,30,20,21,22,29,25)(17,34,37,39,32,40,36,31);(11,22)(12,27)(13,30)(14,26)(16,34)(17,37)(18,33)(20,23)(21,29)(24,40)(25,28)(31,35)(32,36)(38,39)Polycyclic presentation:
2^10|b^2=f,c^2=ghj,d^2=fij,f^2=h,g^2=i,i^2=j,c^b=cg,d^a=dj,d^b=dg,d^c=dgij,e^c=eghj,e^d=eghj,f^c=fj,f^d=fj,g^b=gi,g^d=gi,g^e=gij,i^b=ij,i^d=ij,i^e=ij154 conjugacy classes — view the full character table.
| maximal subgroup of | #72286625, #72287030, #72287099, … |
| maximal normal subgroup of | #72286625, #72287030, #72287099, … |
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, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);