Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 9421 conjugacy classes of subgroups; only the 1253 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,11,12)(17,18,21,22)(19,23,24,20);(2,4)(5,6,7,8)(9,10,11,12)(14,16)(19,24)(20,23);(13,14)(15,16)(18,22)(19,24);(6,8)(9,10,11,12)(17,19,21,24)(18,23,22,20);(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^10|a^2=f,b^2=hj,d^2=f,e^2=fhij,b^a=bfh,c^a=cj,c^b=cg,d^b=dfhj,d^c=dj,e^a=efi,e^b=efi,e^d=ehj136 conjugacy classes — view the full character table.
| maximal subgroup of | #72333212, #72333231, #72333259, … |
| maximal normal subgroup of | #72333212, #72333231, #72333259, … |
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.6);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);