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 11277 conjugacy classes of subgroups; only the 4081 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24);(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24);(1,2)(3,4)(5,6,8,7)(10,14)(11,15)(17,18,21,24)(19,22,23,20);(1,2,3,4)(5,6)(7,8)(10,14)(12,16)(17,19)(18,20)(21,23)(22,24);(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^10|c^2=h,d^2=hij,f^2=h,b^a=bg,c^a=cgi,c^b=cj,d^a=dgi,d^c=dij,e^c=ehj,e^d=ehj,f^a=fgij,f^c=fg,f^d=fi,f^e=fh142 conjugacy classes — view the full character table.
| maximal subgroup of | #72329601 |
| maximal normal subgroup of | #72329601 |
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, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);