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 22553 conjugacy classes of subgroups; only the 29707 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)(3,6)(4,8)(5,7)(11,15)(13,16)(17,18,21,23)(19,22,24,20);(1,3)(2,6)(4,5)(7,8)(11,15)(13,16);(1,2)(3,8)(4,6)(5,7)(11,15)(13,16);(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24);(17,19)(18,22)(20,23)(21,24);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^10|a^2=h,d^2=hj,g^2=i,c^b=ci,e^d=ej,f^d=fh,f^e=fh,g^a=gij,g^b=gij,g^c=gj,g^e=gi196 conjugacy classes — view the full character table.
| maximal subgroup of | #72304523 |
| maximal normal subgroup of | #72304523 |
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.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);