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 ◁ #17323 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2354 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)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,35)(31,41)(32,42)(33,43)(36,48)(37,45)(38,46)(39,44)(40,47);(1,2,4,8,5,6,9,13)(3,7,10,14,11,12,15,16)(17,19,21,20,23,33,38,36,24,31,22,35,39,37,40,32)(18,25,27,34,29,43,46,48,30,41,28,26,44,45,47,42);(2,6)(7,12)(8,13)(14,16)(19,31)(20,35)(25,41)(26,34)(32,36)(33,37)(42,48)(43,45);(17,20,22,33,23,36,40,31,24,35,21,37,39,32,38,19)(18,26,28,45,29,42,47,25,30,34,27,43,44,48,46,41);(1,3,9,15,5,11,4,10)(2,7,13,16,6,12,8,14)(19,32,31,36)(20,33,35,37)(21,38)(22,40)(23,39)(25,42,41,48)(26,45,34,43)(27,46)(28,47)(29,44)Polycyclic presentation:
2^10|b^2=f,d^2=g,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,c^b=chi,d^a=dj,d^b=di,d^c=dj,e^b=eg,e^d=egij,f^d=fj,f^e=fi,g^b=gj,g^e=gi,h^e=hj,i^e=ij124 conjugacy classes — view the full character table.
| maximal subgroup of | #72286877, #72287030, #72318539, … |
| maximal normal subgroup of | #72286877, #72287030, #72318539, … |
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.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.6*F.7*F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);