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 23814 conjugacy classes of subgroups; only the 29439 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,23)(22,24);(1,3,5,6)(2,4,7,8);(1,2)(3,8)(4,6)(5,7)(17,19)(18,21)(20,24)(22,23);(1,3,5,6)(2,4,7,8)(17,18)(19,20)(21,23)(22,24);(9,10,12,14)(11,13,16,15)(17,20,22,21)(18,19,24,23);(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^10|b^2=h,d^2=h,e^2=i,f^2=j,b^a=bh,c^b=ch,d^a=dh,d^c=dhj,e^a=ei,f^a=fj,f^e=fij,g^e=gj,g^f=ghi166 conjugacy classes — view the full character table.
| maximal subgroup of | #72234898 |
| maximal normal subgroup of | #72234898 |
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, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);