Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5168765 |
| 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 | |
3216 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)(5,9)(6,10)(11,25,27,20)(12,28)(13,19)(14,29,17,22)(16,21)(18,24)(26,32,30,31);(7,19)(14,17)(20,25)(21,28)(22,29)(26,30);(1,3)(2,7,10,19)(4,11)(5,16,18,12)(6,13)(8,20,23,25)(9,21)(14,30,17,26)(15,27)(22,31)(24,28)(29,32);(1,4)(2,8,10,23)(3,11)(5,17)(6,15)(7,20,19,25)(9,22,24,29)(12,26)(13,27)(14,18)(16,30)(21,32,28,31);(1,5)(2,9)(3,12)(4,14)(6,18)(7,21)(8,22)(10,24)(11,26)(13,16)(15,17)(19,28)(20,31)(23,29)(25,32)(27,30)Polycyclic presentation:
2^10|a^2=g,c^2=f,d^2=i,b^a=bh,c^a=cghj,c^b=ci,d^a=dghi,d^b=dfi,d^c=dfi,e^a=egh,e^b=eg,e^c=ehi,e^d=ehij,f^a=fj,f^e=fj,g^d=gj,h^c=hj,i^a=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72334872, #72334886, #72334898, … |
| maximal normal subgroup of | #72334872, #72334886, #72334898, … |
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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
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);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
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);