Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #2177 |
| commutator | C2×C16 | C23×C4 |
| Frattini, cosocle | C4×C16 | C24 |
| socle | C22 | C2×#2177 |
| prime radical | C2×#2262 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,21)(12,28)(13,33)(14,36)(15,16)(17,24)(18,31)(19,30)(20,34)(22,45)(23,38)(25,46)(26,59)(27,60)(29,41)(32,65)(35,66)(37,56)(39,63)(40,54)(42,70)(43,50)(44,53)(47,57)(48,55)(49,61)(51,69)(52,68)(58,72)(62,64)(67,71);(1,3,4,7,5,8,2,6)(11,22,25,27)(12,29)(17,40,37,43)(18,38)(21,45,46,60)(23,31)(24,54,56,50)(26,51,58,55)(28,41)(32,64)(42,68,67,47)(44,61)(48,59,69,72)(49,53)(52,71,57,70)(62,65);(10,16)(17,37)(18,38)(19,39)(20,36)(21,46)(28,41)(33,66)(40,43)(42,67)(44,61)(45,60)(47,68)(48,69)(59,72)(62,65);(9,11,23,52,13,26,32,54,34,22,49,70,63,51,12,24,15,25,31,57,35,58,64,50,14,27,53,71,30,55,29,56)(10,17,41,69,19,42,44,45,36,40,62,59,66,68,18,21,16,37,28,48,39,67,61,60,20,43,65,72,33,47,38,46)Polycyclic presentation:
2^10|a^2=h,b^2=f,d^2=egj,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=cj,d^a=de,d^b=dij,e^a=eg,e^b=ej,f^d=fj,g^a=gi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72234370, #72321149, #72321160, … |
| maximal normal subgroup of | #72234370, #72321149, #72321160, … |
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, 2, F.6);;
gap> SetPower(coll, 4, F.5*F.7*F.10);;
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, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*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);