Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #499832 |
| commutator, Frattini, cosocle | #532 | C24 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2712 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)(3,12)(4,9)(5,21)(6,10)(7,11)(8,15)(13,24)(14,27)(16,26)(17,25)(18,32)(19,29)(20,30)(22,28)(23,31);(1,3,5,16,7,17,6,15)(2,8,21,12,11,26,10,25)(4,18,22,14,20,13,19,31)(9,27,28,24,30,23,29,32);(3,13)(8,23)(12,24)(14,16)(15,31)(17,18)(25,32)(26,27);(1,4,6,19,7,20,5,22)(2,9,10,29,11,30,21,28)(3,14,16,13,17,31,15,18)(8,24,12,23,26,32,25,27)Polycyclic presentation:
2^10|b^2=hij,d^2=h,f^2=j,g^2=i,h^2=j,b^a=bf,c^b=ce,d^a=di,d^b=dhj,d^c=dg,e^b=ei,e^d=eh,f^a=fj,f^b=fj,f^d=fj,g^b=gh,g^c=gi,g^e=gi,h^b=hj,h^c=hi,h^e=hj,i^b=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72339877, #72339883, #72339887, … |
| maximal normal subgroup of | #72339877, #72339883, #72339887, … |
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.8*F.9*F.10);;
gap> SetPower(coll, 4, 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, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);