Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2760 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #233549 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,34)(28,43)(29,36)(30,37)(31,38)(32,39)(33,40)(35,45)(41,48)(42,47)(44,46);(1,3)(2,6,8,9)(4,12)(5,11)(7,15,16,14)(10,13)(17,19)(18,23)(20,27,31,42)(21,28)(22,29)(24,34,38,47)(25,35)(26,36)(30,41,32,44)(33,45)(37,46,39,48)(40,43);(1,4)(2,7)(3,10,11,12)(5,13)(6,14,9,15)(8,16)(17,20)(18,24)(19,27)(21,32,33,30)(22,31)(23,34)(25,39,40,37)(26,38)(28,44,45,41)(29,42)(35,48,43,46)(36,47);(7,16)(14,15)(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,37)(26,40)(27,41)(29,45)(31,32)(34,46)(36,43)(38,39)(42,44)(47,48);(2,8)(4,13)(6,9)(10,12)(18,26)(23,36)(24,38)(25,40)(34,47)(35,43)(37,39)(46,48)Polycyclic presentation:
2^10|b^2=f,c^2=h,b^a=bh,c^b=cg,d^a=dgi,d^c=df,e^a=eij,e^c=ej,f^a=fj,f^c=fi,g^b=gi,g^c=gj,h^b=hj,h^d=hi70 conjugacy classes — view the full character table.
| maximal subgroup of | #72284709, #72340246, #72340356, … |
| maximal normal subgroup of | #72284709, #72340246, #72340356, … |
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.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);