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 | #1442757 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | C22×B(4,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3770 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,26,16,19)(13,20,29,23)(15,22)(17,24)(18,25)(21,30)(27,32,28,31);(4,15)(12,16)(21,24)(31,32);(1,3)(2,7)(4,16)(5,13,18,29)(6,14)(8,19)(9,23,25,20)(10,11)(12,15)(17,28,30,27)(21,32,24,31)(22,26);(1,4,6,15)(2,8)(3,12,14,16)(5,17,18,30)(7,19)(9,24)(10,22)(11,26)(13,28,29,27)(20,31)(21,25)(23,32);(1,5)(2,9)(3,13)(4,17)(6,18)(7,20)(8,21)(10,25)(11,23)(12,27)(14,29)(15,30)(16,28)(19,31)(22,24)(26,32)Polycyclic presentation:
2^10|a^2=h,c^2=f,d^2=ij,b^a=bg,c^a=cgi,d^a=dgj,d^b=dfij,d^c=dfij,e^a=egh,e^b=eg,e^c=ehij,e^d=ehi,f^e=fj,g^d=gj,h^c=hj,i^a=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72334823, #72334827, #72334833, … |
| maximal normal subgroup of | #72334823, #72334827, #72334833, … |
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, 3, F.6);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, 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);