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 7131 conjugacy classes of subgroups; only the 3761 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(17,18)(19,24)(20,22)(21,23);(6,8)(10,13)(14,16);(5,6)(7,8)(17,19,21,20)(18,22,23,24);(2,4)(10,13)(11,15)(17,18)(19,22)(20,24)(21,23);(1,2,3,4)(9,10,12,13)(11,14,15,16);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^10|c^2=h,e^2=hi,f^2=gh,c^a=ch,c^b=cj,e^b=eg,e^d=ehi,f^d=fi,f^e=fghi154 conjugacy classes — view the full character table.
| maximal subgroup of | #72323339, #72323383, C2×#72323299, … |
| maximal normal subgroup of | #72323339, #72323383, C2×#72323299, … |
| maximal quotient of | #72323727, #72323730, #72331105, … |
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, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);