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 | #7624074 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17330 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2265 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)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(19,29)(20,34)(24,36)(25,31)(30,38)(33,35)(42,48)(44,47);(1,3,6,10)(2,7,9,5)(4,12,14,11)(8,13,16,15)(19,30)(20,35)(21,32)(22,40)(24,42)(25,44)(26,43)(27,41)(29,38)(31,47)(33,34)(36,48)(37,46)(39,45);(1,4,5,13,6,14,7,15)(2,8,10,12,9,16,3,11)(17,18)(19,31,29,25)(20,36,34,24)(21,39)(22,41)(23,28)(26,46)(27,40)(30,47,38,44)(32,45)(33,42,35,48)(37,43);(17,19,32,20,22,33,37,38,23,29,46,34,40,35,21,30)(18,24,43,31,27,44,45,48,28,36,39,25,41,47,26,42);(2,9)(3,10)(8,16)(11,12)(17,20,37,29,40,30,32,33,23,34,21,19,22,38,46,35)(18,25,45,24,41,48,43,47,28,31,26,36,27,42,39,44)Polycyclic presentation:
2^10|b^2=h,c^2=f,d^2=gi,e^2=gj,f^2=h,g^2=ij,i^2=j,c^b=ch,d^a=dj,d^b=dg,d^c=dgi,e^a=ehj,e^b=eghij,e^c=eg,f^d=fj,f^e=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij103 conjugacy classes — view the full character table.
| maximal subgroup of | #72287243, #72287684, #72318898, … |
| maximal normal subgroup of | #72287243, #72287684, #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, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);