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 | #4894722 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | C24×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3248 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)(4,8,15,23)(5,16)(6,10)(9,18)(11,19,26,31)(12,27)(13,21)(14,24,17,22)(20,29)(25,32,28,30);(1,3)(2,7)(4,11)(5,12,18,29)(6,13)(8,19)(9,20,16,27)(10,21)(14,28,17,25)(15,26)(22,32,24,30)(23,31);(1,4)(2,8,10,23)(3,11)(5,17,18,14)(6,15)(7,19,21,31)(9,24)(12,28,29,25)(13,26)(16,22)(20,32)(27,30);(1,5)(2,9)(3,12)(4,14)(6,18)(7,20)(8,22)(10,16)(11,25)(13,29)(15,17)(19,30)(21,27)(23,24)(26,28)(31,32);(7,21)(19,31)(20,27)(30,32)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=fi,c^a=chi,c^b=cf,d^a=dj,d^b=dg,d^c=dh,e^a=eghj,e^b=ei,f^d=fj,g^b=gj,g^c=gj,h^c=hj,i^a=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72334801, #72334827, #72334861, … |
| maximal normal subgroup of | #72334801, #72334827, #72334861, … |
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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*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, 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);