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 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7623255 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C2×#1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,21,26,20,27,34,42,22,25,35,23,33,37,46,38)(19,28,30,36,31,43,45,47,32,24,39,40,29,41,44,48);(1,2,4,8,5,7,12,15)(3,9,11,16,10,6,13,14)(18,23,25,26)(20,33)(21,34)(24,36,28,40)(27,42,37,38)(29,31)(30,45)(35,46)(39,44)(41,48,43,47);(17,19)(18,24)(20,31)(21,30)(22,32)(23,36)(25,28)(26,40)(27,41)(29,33)(34,45)(35,39)(37,43)(38,47)(42,48)(44,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,13)(15,16)(18,25)(23,26)(24,28)(27,37)(36,40)(38,42)(41,43)(47,48);(2,7)(3,10)(8,15)(11,13)(17,20)(18,26)(19,29)(22,33)(23,25)(24,36)(27,38)(28,40)(30,39)(31,32)(34,46)(37,42)(41,48)(43,47)Polycyclic presentation:
2^10|a^2=g,b^2=f,f^2=h,g^2=i,i^2=j,b^a=bgi,c^a=cj,d^a=dj,d^b=dh,e^a=eg,e^b=ehj,e^c=ej,e^d=eh,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72322319, #72322376, #72322467, … |
| maximal normal subgroup of | #72322319, #72322376, #72322467, … |
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.7);;
gap> SetPower(coll, 2, F.6);;
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.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);