Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13047 |
| commutator | C4⋊Q8 | C23×C4 |
| Frattini, cosocle | #532 | C24 |
| prime radical | #6348449 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,9)(3,14)(5,21)(6,7)(10,25)(11,24)(12,26)(13,22)(15,29)(16,19)(17,18)(23,31)(27,28)(30,32);(1,2,4,11)(3,15,17,29)(5,22,6,12)(7,24,21,9)(8,13,20,26)(10,18,28,14)(16,30,23,32)(19,27,31,25);(1,3,7,19,8,16,6,14)(2,10,12,30,13,27,24,15)(4,18,5,23,20,31,21,17)(9,25,26,32,22,28,11,29);(3,16)(4,20)(5,21)(9,22)(10,27)(11,26)(14,19)(15,30)Polycyclic presentation:
2^10|b^2=e,c^2=h,f^2=j,g^2=i,h^2=j,b^a=bf,c^a=cg,c^b=cghij,d^a=dij,d^b=di,d^c=dj,e^c=eh,e^d=ej,f^a=fj,f^b=fj,f^c=fh,f^d=fj,g^a=gi,g^b=gh,g^e=gi,g^f=gi,h^a=hi,h^b=hij,h^e=hj,h^f=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72339824, #72339863, #72339867, … |
| maximal normal subgroup of | #72339824, #72339863, #72339867, … |
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.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, 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, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);