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 | |
2188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(6,10)(11,15)(13,16)(17,18)(19,24)(20,32)(21,27)(22,28)(23,25)(26,29)(30,38)(31,39)(33,41)(34,48)(35,40)(36,37)(42,46)(43,47)(44,45);(1,2)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,24)(20,32)(21,27)(22,28)(23,25)(26,29)(30,38)(31,39)(33,41)(34,48)(35,40)(36,37)(42,46)(43,47)(44,45);(3,9)(6,10)(11,15)(13,16)(17,19,21,20,22,31,33,34,23,30,35,29,36,46,45,47)(18,24,27,32,28,39,41,48,25,38,40,26,37,42,44,43);(1,3,4,11,5,9,12,15)(2,6,8,13,7,10,14,16)(18,25)(19,29,30,20)(21,33)(22,36)(24,32,38,26)(27,44)(31,34,46,47)(35,45)(39,43,42,48)(40,41);(2,7)(6,10)(8,14)(13,16)(17,20,33,30,36,47,21,31,23,29,45,19,22,34,35,46)(18,26,41,24,37,48,27,42,25,32,44,38,28,43,40,39)Polycyclic presentation:
2^10|c^2=g,d^2=f,e^2=gi,f^2=h,g^2=i,i^2=j,d^a=dhj,d^b=dhj,d^c=dghi,e^a=ej,e^b=ehj,e^d=egij,f^c=fj,f^e=fj,g^d=gi,i^d=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72287431, #72287615, #72287644, … |
| maximal normal subgroup of | #72287431, #72287615, #72287644, … |
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, 3, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);