Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C25 |
| prime radical | C22×#453 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15,21,28)(12,16)(13,17)(14,18)(19,33,35,26)(20,36,38,27)(22,29)(23,30)(24,31)(25,32)(34,39,37,40);(11,19)(13,24)(15,26)(17,31)(20,37)(21,35)(23,25)(27,40)(28,33)(30,32)(34,38)(36,39);(3,8)(5,7)(9,11,14,21)(10,15)(12,20)(13,19)(16,27,29,36)(17,26,31,33)(18,28)(22,38)(23,37,25,34)(24,35)(30,40)(32,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,20)(13,25)(14,22)(15,27)(17,32)(18,29)(19,34)(21,38)(23,24)(26,39)(28,36)(30,31)(33,40)(35,37);(12,22)(16,29)(20,38)(23,25)(27,36)(30,32)(34,37)(39,40)Polycyclic presentation:
2^10|a^2=f,c^2=hij,d^2=fi,g^2=j,b^a=bi,c^a=ch,c^b=cg,d^a=df,d^b=di,d^c=dfj,e^d=ej,f^c=fj,g^a=gi,g^b=gj,g^c=gij,g^d=gj,h^a=hj,h^b=hi,h^c=hj,h^d=hj,h^g=hj,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72318209, #72318803, #72318828, … |
| maximal normal subgroup of | #72318209, #72318803, #72318828, … |
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.6);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);