Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#386 |
| commutator | C22×C4 | C23×C8 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| socle | C22 | C2×#1869 |
| prime radical | C22×#127 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,12)(7,14)(9,15)(10,16);(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^10|b^2=f,c^2=g,e^2=h,g^2=i,i^2=j,b^a=bf,c^b=ce,d^c=dj,e^a=eh,f^c=fh,g^b=gh160 conjugacy classes — view the full character table.
| maximal subgroup of | #72239332, #72239340, #72239383, … |
| maximal normal subgroup of | #72239332, #72239340, #72239383, … |
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, 3, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);