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 | #7623529 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17336 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,9)(8,12)(11,16)(17,18)(19,25)(20,31)(21,32)(22,28)(23,29)(24,26)(27,37)(30,43)(33,42)(34,38)(35,44)(36,46)(39,48)(40,47)(41,45);(1,2,5,9,6,7,10,3)(4,11,14,12,15,16,13,8)(17,19,23,27,24,33,41,44,22,30,40,32,20,34,36,48)(18,25,29,37,26,42,45,35,28,43,47,21,31,38,46,39);(1,2,5,9,6,7,10,3)(4,8,13,16,15,12,14,11)(17,20,22,24)(18,26,28,31)(19,30)(21,37)(23,41,40,36)(29,46,47,45)(33,34)(35,39);(1,3,10,7,6,9,5,2)(4,12,13,11,15,8,14,16)(17,21,36,43,20,35,40,42,22,37,41,25,24,39,23,38)(18,27,46,19,31,48,47,34,28,32,45,30,26,44,29,33);(1,4,10,14,6,15,5,13)(2,8,3,11,7,12,9,16)(18,28)(19,32,30,27)(20,24)(21,25,37,43)(23,41)(29,46)(33,44,34,48)(35,42,39,38)(36,40)(45,47)Polycyclic presentation:
2^10|b^2=g,c^2=f,d^2=gij,e^2=fhj,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=chj,c^b=chi,d^a=dhj,d^c=dhij,e^a=ej,e^b=eghi,e^c=ej,e^d=egj,f^b=fj,f^d=fj,g^c=gj,g^e=gi,i^e=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72287517, #72287659, #72318841, … |
| maximal normal subgroup of | #72287517, #72287659, #72318841, … |
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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6*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, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);