Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×#106 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #3444 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ #60204 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(19,27)(23,29)(28,42)(31,43)(32,44)(37,45)(38,47)(39,48);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,18,20,24)(19,28)(21,35,33,46)(22,36,34,26)(23,37)(25,41,40,30)(27,42)(29,45)(31,39)(32,47)(38,44)(43,48);(3,10)(6,9)(11,16)(13,15)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,37)(30,35)(31,47)(32,48)(33,40)(34,36)(38,43)(39,44)(41,46)(42,45);(2,7)(3,10)(8,14)(11,16)(17,19,30,37,34,44,21,31,20,27,41,45,22,32,33,43)(18,23,35,42,36,48,25,38,24,29,46,28,26,39,40,47);(2,7)(6,9)(8,14)(13,15)(18,24)(19,27)(25,40)(26,36)(31,43)(32,44)(35,46)(37,45)Polycyclic presentation:
2^10|b^2=f,d^2=ghj,f^2=h,g^2=i,i^2=j,b^a=bh,c^b=chj,d^a=dhj,d^b=dg,d^c=dj,e^a=eh,e^b=ej,e^c=ej,e^d=ej,f^d=fj,g^b=gi,i^b=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72287669, #72287678, #72287792, … |
| maximal normal subgroup of | #72287669, #72287678, #72287792, … |
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, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);