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 | #21503 |
| commutator | C4×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| prime radical | #490756 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #7846 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(2,6,8,15)(3,9)(4,12)(7,10,13,16)(11,14)(17,18,22,28,24,29,23,39)(19,25,32,41,34,42,33,47)(20,26,35,43,36,27,21,37)(30,40,31,46,45,44,38,48);(1,2)(3,10)(4,13)(5,8)(6,14)(7,12)(9,15)(11,16)(17,19)(18,25)(20,30)(21,38)(22,32)(23,33)(24,34)(26,40)(27,44)(28,41)(29,42)(31,35)(36,45)(37,48)(39,47)(43,46);(2,7,8,13)(3,11)(6,10,15,16)(9,14)(17,20,23,21,24,36,22,35)(18,26,39,37,29,27,28,43)(19,30,33,38,34,45,32,31)(25,40,47,48,42,44,41,46);(17,21,23,36,24,35,22,20)(18,27,28,37,29,26,39,43)(19,31,33,30,34,38,32,45)(25,40,41,46,42,44,47,48)Polycyclic presentation:
2^10|a^2=e,c^2=ei,d^2=g,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bf,c^b=chj,d^a=dg,d^b=di,d^c=di,e^b=ej,f^a=fh,f^b=fhj,f^c=fj,g^a=gi,h^a=hj,h^b=hj82 conjugacy classes — view the full character table.
| maximal subgroup of | #72319854, #72320455, #72320616, … |
| maximal normal subgroup of | #72319854, #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, 1, F.5);;
gap> SetPower(coll, 3, F.5*F.9);;
gap> SetPower(coll, 4, F.7);;
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, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
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.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);