Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3480982 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #34980 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3591 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)(5,17)(6,10)(8,16)(9,19)(11,26)(12,24)(13,22)(15,25)(18,21)(20,28)(23,29)(27,32)(30,31);(1,3)(2,7)(4,11)(5,18,19,12)(6,13)(8,20)(9,24,17,21)(10,22)(14,26)(15,27,29,30)(16,28)(23,31,25,32);(1,4)(2,8,10,14)(3,11)(5,15,19,29)(6,16)(7,20,22,26)(9,25)(12,27,18,30)(13,28)(17,23)(21,32)(24,31);(1,5,6,19)(2,9,10,17)(3,12,13,18)(4,15,16,29)(7,21,22,24)(8,23,14,25)(11,27,28,30)(20,31,26,32);(7,22)(20,26)(21,24)(31,32)Polycyclic presentation:
2^10|b^2=f,c^2=fi,d^2=j,c^a=ch,c^b=cf,d^a=dhj,d^b=dg,d^c=dhij,e^a=egh,e^b=ei,f^d=fj,g^b=gj,g^c=gj,h^c=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72334801, #72334815, #72334880, … |
| maximal normal subgroup of | #72334801, #72334815, #72334880, … |
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.6*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);