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 | |
74694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(6,7)(10,11)(13,14)(15,19)(16,18)(17,20);(2,4)(5,6)(7,8)(9,10,12,11)(13,15,17,16)(14,18,20,19);(10,11)(13,14)(15,18)(16,19)(17,20);(2,4)(5,6,8,7)(9,11,12,10)(13,14,17,20)(15,18,16,19);(6,7)(13,14)(15,19)(16,18)(17,20);(1,2,3,4)(5,7,8,6)(13,16,17,15)(14,19,20,18)Polycyclic presentation:
2^10|b^2=ij,d^2=i,f^2=ghij,b^a=bi,c^b=ch,d^a=di,d^b=dj,d^c=dij,e^b=ehi,e^d=ehi,f^a=fhi,f^b=fg,f^d=fgj,f^e=fhi190 conjugacy classes — view the full character table.
| maximal subgroup of | #72233580, #72298072, C2×#72233570, … |
| maximal normal subgroup of | #72233580, #72298072, C2×#72233570, … |
| maximal quotient of | #72298590, #72298604, #72298624, … |
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.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);