Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C26 |
| commutator | C23 | C25×C4 |
| prime radical | C24×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
80173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,15)(14,16)(17,18)(19,20)(21,23)(22,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(1,2,3,4)(6,8)(10,13)(14,16)(18,23)(19,24);(2,4)(5,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,19,23,24);(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^10|c^2=h,d^2=ij,b^a=bg,c^a=cg,c^b=cj,d^a=dgj,d^c=dhi,e^c=eg,e^d=eg,f^c=fhij,f^d=fhij208 conjugacy classes — view the full character table.
| maximal subgroup of | #72334342, #72334357, C2×#72334336, … |
| maximal normal subgroup of | #72334342, #72334357, C2×#72334336, … |
| maximal quotient of | #72334417, C2×#72334336, #72335156, … |
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.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);