Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #115134 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C24×D4 | C23 |
| prime radical | C2×#3484 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
3429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,13,24)(4,15)(5,18,20,10)(6,9)(7,14)(8,16)(12,25)(17,30,31,23)(19,22)(21,28,29,27)(26,32);(1,3)(2,7)(4,12)(5,19)(6,13)(8,21)(9,22)(10,24)(11,18)(14,20)(15,29)(16,26)(17,28)(23,25)(27,31)(30,32);(1,4,5,17)(2,8,10,23)(3,12,14,28)(6,16,20,31)(7,21,24,32)(9,15,18,30)(11,25,22,29)(13,26,19,27)Polycyclic presentation:
2^10|a^2=d,c^2=defghj,g^2=j,b^a=bg,c^a=cdf,c^b=cde,d^b=di,d^c=dh,e^a=eh,e^b=ei,e^c=ehi,f^a=fh,f^b=fi,f^c=fhij,g^a=gij,g^b=gj,g^c=gh,g^d=gj,h^b=hj,h^c=hj,i^a=ij,i^c=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72255981, #72256047, #72256091, … |
| maximal normal subgroup of | #72255981, #72256047, #72256091, … |
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.4);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);