Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #15947 |
| commutator, Frattini, cosocle | C24×C4 | C24 |
| LCS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
5523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(1,3)(2,7)(4,6)(5,12,13,11)(8,10)(9,15,16,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,14,16,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(22,31)(25,32)Polycyclic presentation:
2^10|b^2=h,c^2=i,f^2=j,b^a=bh,c^b=cf,d^a=dg,d^b=dg,d^c=de,e^a=ei,e^b=ej,f^b=fj,f^c=fj,f^d=fij,g^b=gij,g^c=gi,h^d=hij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72312782, #72312812, #72312857, … |
| maximal normal subgroup of | #72312782, #72312812, #72312857, … |
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);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);