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 | |
71531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6,7,8)(10,12)(14,17)(18,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(15,19)(18,20);(1,2)(3,4)(10,12)(13,14)(15,20)(16,17)(18,19);(5,6)(7,8)(9,10)(11,12);(1,2,3,4)(5,6,7,8)(13,15)(14,18)(16,19)(17,20);(1,2)(3,4)(5,6)(7,8)(13,14)(15,20)(16,17)(18,19)Polycyclic presentation:
2^10|a^2=g,e^2=g,b^a=bgj,c^a=ci,c^b=cj,d^a=dh,d^c=dj,e^b=ehij,e^c=ei,e^d=ehj,f^a=fhij,f^e=fhij178 conjugacy classes — view the full character table.
| maximal subgroup of | #72335137, C2×#72335133, #72337228 |
| maximal normal subgroup of | #72335137, C2×#72335133, #72337228 |
| maximal quotient of | #72335158, C2×#72335133, #72337244, … |
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.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);