Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2199533 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #34980 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,14)(5,9)(6,10)(8,16)(11,27)(12,21)(13,22)(15,23)(17,24)(18,25)(19,26)(20,29)(28,31)(30,32);(1,3,6,13)(2,7,10,22)(4,11,16,29)(5,17)(8,20,14,27)(9,24)(12,19)(15,28)(18,30)(21,26)(23,31)(25,32);(1,4)(2,8,10,14)(3,11)(5,18)(6,16)(7,20,22,27)(9,25,26,23)(12,28)(13,29)(15,19)(17,30)(21,31,24,32);(1,5)(2,9)(3,12)(4,15)(6,19)(7,21)(8,23)(10,26)(11,28)(13,17)(14,25)(16,18)(20,31)(22,24)(27,32)(29,30);(7,22)(20,27)(21,24)(31,32)Polycyclic presentation:
2^10|b^2=f,c^2=i,c^a=cfg,d^a=dgj,d^b=dgh,d^c=dh,e^a=ehj,e^b=ei,f^d=fj,g^c=gj,h^b=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72317935, #72319168, #72319229, … |
| maximal normal subgroup of | #72317935, #72319168, #72319229, … |
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, 2, F.6);;
gap> SetPower(coll, 3, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);