Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
166833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(13,14)(15,18)(16,20)(17,19);(6,8);(2,4)(6,8)(9,10,11,12)(13,15,17,20)(14,18,19,16);(6,8)(15,20)(16,18);(2,4)(6,8)(13,14)(15,18)(16,20)(17,19);(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^10|c^2=g,f^2=ghi,d^c=dgi,f^a=fgi,f^b=fh,f^c=fij,f^d=fghi,f^e=fj208 conjugacy classes — view the full character table.
| maximal subgroup of | #72256579, #72268589, C2×#72256574 |
| maximal normal subgroup of | #72256579, #72268589, C2×#72256574 |
| maximal quotient of | #72286609, C2×#72256574, #72287891, … |
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, 3, F.7);;
gap> SetPower(coll, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);