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 | #7624015 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17320 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2228 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)(8,11)(9,14)(10,16)(17,18,22,30,23,31,35,44,24,29,40,27,41,45,42,47)(19,25,32,36,26,43,46,48,33,21,39,20,34,37,28,38);(1,2)(3,8)(4,14)(5,10)(6,7)(9,12)(11,13)(15,16)(17,19)(18,25)(20,27)(21,29)(22,32)(23,26)(24,33)(28,42)(30,36)(31,43)(34,41)(35,46)(37,45)(38,47)(39,40)(44,48);(1,2)(3,11)(4,9)(5,10)(6,7)(8,13)(12,14)(15,16)(17,20,35,25,41,48,22,37,24,36,42,21,23,38,40,43)(18,26,44,39,45,19,30,46,29,34,47,32,31,33,27,28);(1,3,5,12,6,13,15,4)(2,8,10,9,7,11,16,14)(18,27,29,30)(20,21,36,25)(22,35)(23,41)(26,34)(28,39)(31,44,45,47)(32,46)(37,38,43,48)(40,42);(1,4,15,13,6,12,5,3)(2,9,16,8,7,14,10,11)(17,21,35,48,41,43,22,36,24,25,42,38,23,37,40,20)(18,28,47,26,45,39,27,19,29,46,44,34,31,32,30,33)Polycyclic presentation:
2^10|a^2=g,c^2=gi,d^2=f,e^2=fghi,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=chj,c^b=cj,d^a=dgi,d^c=dgh,e^a=eij,e^b=eh,e^c=ei,e^d=egj,f^a=fj,f^c=fj,f^e=fj,g^d=gi,g^e=gj,i^d=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72287829, #72318841, #72318898, … |
| maximal normal subgroup of | #72287829, #72318841, #72318898, … |
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, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9);;
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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*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, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);