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 | C22×C8 | C25 |
| prime radical | #7623723 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ #20493 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
2687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(12,23)(18,31)(21,28)(24,40)(25,39)(26,35)(33,37)(36,38);(1,2,4,6)(3,5,7,8)(9,10,14,16)(11,20,15,17)(12,24,23,40)(13,29,30,34)(18,35,31,26)(19,22,32,27)(21,38,28,36)(25,33,39,37);(10,16)(11,15)(12,25,23,39)(13,22)(18,36,31,38)(19,29)(21,35,28,26)(24,37,40,33)(27,30)(32,34);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,26,23,35)(13,30)(14,15)(16,20)(18,24,31,40)(19,32)(21,25,28,39)(33,38,37,36);(2,6)(5,8)(9,12,27,25,15,28,30,35,14,23,22,39,11,21,13,26)(10,18,29,38,20,37,32,40,16,31,34,36,17,33,19,24)Polycyclic presentation:
2^10|b^2=h,c^2=f,d^2=fhj,e^2=ghij,g^2=ij,i^2=j,c^b=cj,d^a=dhj,d^b=dj,d^c=dj,e^a=ej,e^b=eg,e^c=eghij,e^d=eghij,f^e=fj,g^b=gi,g^c=gij,g^d=gij,i^b=ij,i^c=ij,i^d=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72314251, #72314408, #72314441, … |
| maximal normal subgroup of | #72314251, #72314408, #72314441, … |
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.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);