Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #32898 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C24 |
| prime radical | #417824 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
4285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,19,22,34)(12,16)(13,17)(14,18)(15,21,30,26)(20,31,38,37)(23,29,39,40)(24,32)(25,33)(27,35)(28,36);(2,5)(3,8)(6,7)(9,11)(10,15)(12,23)(13,26)(14,22)(16,31)(17,34)(18,30)(19,36)(20,25)(21,28)(24,39)(27,38)(29,35)(32,37)(33,40);(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,35)(18,32)(19,37)(21,23)(22,38)(25,28)(26,39)(30,40)(31,34)(33,36);(3,8)(5,6)(12,24)(16,32)(25,27)(33,35)Polycyclic presentation:
2^10|a^2=hij,e^2=ij,g^2=i,b^a=be,c^a=chj,c^b=cegi,d^b=df,d^c=dh,e^a=ei,e^b=eij,e^c=eh,e^d=ei,f^a=fi,f^c=fj,g^a=ghi,g^b=gij,g^c=gh,g^d=gi,g^e=gj,h^b=hj67 conjugacy classes — view the full character table.
| maximal subgroup of | #72341048 |
| maximal normal subgroup of | #72341048 |
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.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);