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 | #7624012 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17330 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4,7,9,3,10)(5,13,14,16,15,8,11,12)(17,18,20,26,22,25,30,28)(19,24,32,43,31,42,29,44)(21,35,34,27,38,23,36,40)(33,39,47,41,37,48,46,45);(1,3,7,6)(2,4,9,10)(5,14,15,11)(8,16,13,12)(18,23)(19,29)(20,30)(21,36)(24,41)(25,35)(26,27)(28,40)(31,32)(33,37)(34,38)(39,43)(42,45)(44,48);(1,4,6,9,7,10,3,2)(5,12,14,13,15,16,11,8)(17,19,30,29,22,31,20,32)(18,24,28,44,25,42,26,43)(21,37,36,47,38,33,34,46)(23,39,27,45,35,48,40,41);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(19,31)(24,42)(29,32)(33,37)(39,48)(41,45)(43,44)(46,47);(5,15)(8,13)(11,14)(12,16)(18,25)(23,35)(24,42)(26,28)(27,40)(39,48)(41,45)(43,44)Polycyclic presentation:
2^10|a^2=f,b^2=hj,c^2=fj,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=chj,c^b=chi,d^a=dhj,d^b=dhj,d^c=dj,e^a=ej,e^d=ehj,f^b=fj,g^a=gi,g^b=gij,g^c=gj,i^a=ij,i^b=ij103 conjugacy classes — view the full character table.
| maximal subgroup of | #72318707, #72318841, #72319058, … |
| maximal normal subgroup of | #72318707, #72318841, #72319058, … |
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.6);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.6*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.7);;
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, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);