Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
At least 12958 conjugacy classes of subgroups; only the 4061 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(5,6)(7,8);(2,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18);(5,6,7,8)(9,10)(11,12)(14,18)(15,20);(2,4)(6,8)(9,10)(11,12)(15,20)(16,19);(10,12)(13,15)(14,16)(17,20)(18,19);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^10|b^2=i,c^2=i,b^a=bi,c^a=ci,c^b=cg,d^a=di,d^b=di,d^c=di,e^b=egh,e^c=egh,e^d=egh,f^b=fj,f^c=fi,f^d=fj,f^e=fh154 conjugacy classes — view the full character table.
| maximal subgroup of | #72232191, #72232438, #72263416, … |
| maximal normal subgroup of | #72232191, #72232438, #72263416, … |
| maximal quotient of | #72298570, #72298572, C2×#72232085, … |
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);;
gap> SetPower(coll, 3, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);