Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 7482 conjugacy classes of subgroups; only the 29489 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(18,21)(22,24);(1,2,4,6)(3,5,8,7)(19,23)(22,24);(17,18)(19,22)(20,21)(23,24);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,11,12,16)(10,14,13,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^10|b^2=h,g^2=j,b^a=bh,c^a=chi,d^a=dh,e^b=ehi,e^d=ej,f^b=fi,f^d=fj,g^a=gi,g^f=gj172 conjugacy classes — view the full character table.
| maximal subgroup of | #72256227, #72256415, #72263046 |
| maximal normal subgroup of | #72256227, #72256415, #72263046 |
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.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);