Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C2×#14716 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
22111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(10,12)(21,22,26,28)(23,25,24,27);(2,4)(6,8)(9,10)(11,12)(13,14,16,18)(15,19,20,17)(21,23,26,24)(22,27,28,25);(6,8)(15,20)(17,19)(23,24)(25,27);(1,2)(3,4)(5,6,7,8)(13,14)(15,19)(16,18)(17,20)(21,24,26,23)(22,25,28,27);(13,15)(14,17)(16,20)(18,19)(21,25)(22,24)(23,28)(26,27)Polycyclic presentation:
2^10|a^2=i,b^2=f,d^2=ghi,b^a=bj,c^b=ci,d^a=dg,d^b=dfh,d^c=dghi,e^b=ef,e^c=ef,e^d=ef148 conjugacy classes — view the full character table.
| maximal subgroup of | #72333204, #72333218, #72333238, … |
| maximal normal subgroup of | #72333204, #72333218, #72333238, … |
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.9);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);