Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D8 |
| commutator | C8 | C24×C8 |
| Frattini, cosocle | C4×C8 | C25 |
| socle | C22 | C22×C4×D8 |
| prime radical | C2×#3415 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ C8×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,8,7)(11,19)(12,21)(13,23)(16,28)(17,29)(18,27)(20,31)(22,35)(24,34)(25,36)(26,37)(30,40)(32,33)(38,39);(9,10)(11,16)(12,17)(13,18)(14,15)(19,28)(20,25)(21,29)(22,30)(23,27)(24,26)(31,36)(32,38)(33,39)(34,37)(35,40);(10,15)(16,25)(17,26)(18,27)(28,36)(29,37)(30,38)(39,40);(1,3,4,7,5,8,2,6)(11,20)(16,25)(19,31)(22,32)(28,36)(30,38)(33,35)(39,40);(9,11,21,33,13,22,24,31,14,20,34,35,23,32,12,19)(10,16,29,39,18,30,26,36,15,25,37,40,27,38,17,28)Polycyclic presentation:
2^10|a^2=h,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,c^b=cj,e^a=eg,e^d=ej,g^a=gi,i^a=ij232 conjugacy classes — view the full character table.
| maximal subgroup of | #72232013, #72260420, #72260475, … |
| maximal normal subgroup of | #72232013, #72260420, #72260475, … |
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.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
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, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);