Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13348 |
| commutator, Frattini, cosocle | #532 | C24 |
| prime radical | C2×#33559 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(7,22)(8,13)(10,28)(14,31)(15,29)(16,18)(17,20)(19,21)(23,24)(25,32)(26,30);(1,2)(3,13,14,30,16,23,15,32)(4,17,5,20)(6,11)(7,10)(8,18,25,31,24,12,26,29)(9,21,27,19)(22,28);(1,3,6,16)(2,8,11,24)(4,18,5,12)(7,15,22,14)(9,13,27,23)(10,26,28,25)(17,32,20,30)(19,31,21,29);(1,4)(2,9)(3,12)(5,6)(7,21)(8,23)(10,17)(11,27)(13,24)(14,29)(15,31)(16,18)(19,22)(20,28)(25,32)(26,30)Polycyclic presentation:
2^10|b^2=g,c^2=j,g^2=ij,h^2=j,b^a=bg,c^a=ce,c^b=cef,d^b=dhj,d^c=dj,e^b=eh,f^a=fij,f^b=fi,f^c=fj,f^d=fj,f^e=fj,g^a=gij,g^c=ghi,g^d=gi,g^e=gi,h^a=hi,h^b=hij,h^d=hj,h^e=hj,i^c=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72279639, #72286762, #72286910, … |
| maximal normal subgroup of | #72279639, #72286762, #72286910, … |
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, 2, F.7);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);