Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 11815 conjugacy classes of subgroups; only the 3693 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(5,6)(7,8)(10,11)(14,15)(17,18)(19,22)(20,23)(21,24);(2,4)(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,19,20,24)(18,21,23,22);(12,16)(14,15)(19,24)(21,22);(2,4)(5,6)(7,8)(10,11)(12,16)(19,24)(21,22);(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,2)(3,4)(5,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^10|b^2=i,e^2=gh,b^a=bi,c^b=cg,d^b=di,e^a=egh,e^b=eghi,e^d=egh,f^b=fj,f^c=fi,f^d=fj,f^e=fhi142 conjugacy classes — view the full character table.
| maximal subgroup of | #72232191, #72232371, #72263231, … |
| maximal normal subgroup of | #72232191, #72232371, #72263231, … |
| maximal quotient of | C2×#72232097, #72330706, #72331180, … |
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, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
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*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);