Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #22858 |
| commutator | C4×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| prime radical | #489283 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #8192 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(17,18)(19,23)(20,24)(21,25)(22,27)(26,33)(28,34)(29,37)(30,36)(31,40)(32,39)(35,43)(38,44)(41,45)(42,47)(46,48);(2,6,8,15)(3,9)(4,12)(7,10,13,16)(11,14)(17,19,21,30,22,29,33,43)(18,23,25,36,27,37,26,35)(20,28,32,41,31,42,44,48)(24,34,39,45,40,47,38,46);(1,2)(3,10)(4,13)(5,8)(6,14)(7,12)(9,15)(11,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,34)(25,39)(26,38)(27,40)(29,42)(30,41)(33,44)(35,46)(36,45)(37,47)(43,48);(2,7,8,13)(3,11)(6,10,15,16)(9,14)(18,25,27,26)(19,29)(20,31)(23,35,37,36)(24,38,40,39)(30,43)(32,44)(34,45,47,46)Polycyclic presentation:
2^10|b^2=e,d^2=egi,e^2=i,f^2=h,g^2=i,h^2=j,c^b=cf,d^a=dg,d^b=di,d^c=dhij,e^c=ej,f^b=fh,f^c=fhj,f^d=fj,g^a=gi,h^b=hj,h^c=hj73 conjugacy classes — view the full character table.
| maximal subgroup of | #72320294, #72320455, #72320616, … |
| maximal normal subgroup of | #72320294, #72320455, #72320616, … |
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.5);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);