Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #405104 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| prime radical | C2×#23895 | C2 |
| LCS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
5479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,12,21)(4,14)(5,10,19,26)(6,9)(8,17,24,16)(11,27)(13,31)(15,32,29,20)(18,25)(22,30)(23,28);(1,3)(2,7,9,21)(4,15,18,29)(5,13)(6,12)(8,23)(10,22,26,31)(11,16,28,17)(14,20)(19,30)(24,27)(25,32);(1,4,6,18)(2,8)(3,11,12,28)(5,17,19,16)(7,20)(9,24)(10,14)(13,29,30,15)(21,32)(22,27)(23,31)(25,26);(2,9)(10,26)(11,28)(15,29)Polycyclic presentation:
2^10|a^2=e,b^2=h,c^2=fgh,f^2=j,g^2=j,b^a=beh,c^a=cg,c^b=cf,d^a=dhij,d^b=dh,d^c=di,e^b=ej,e^c=ei,f^a=fij,f^c=fj,f^e=fj,g^a=gij,g^e=gj,h^a=hj,h^c=hj,i^a=ij,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72256030, #72256074, #72256091, … |
| maximal normal subgroup of | #72256030, #72256074, #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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, 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, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);