Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | #7622822 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ #20384 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,31)(20,34)(21,32)(22,33)(23,29)(24,25)(26,35)(27,36)(28,30)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(9,10)(11,19,23,34)(12,24)(13,18)(14,15)(16,31,29,20)(17,25)(21,37,27,38)(22,35)(26,33)(28,30)(32,40,36,39);(1,4,5,8)(2,6,3,7)(10,15)(11,20,23,31)(12,25)(16,19,29,34)(18,30)(21,39,27,40)(22,26)(32,37,36,38);(2,3)(6,7)(11,21,23,27)(12,26)(13,28)(16,32,29,36)(17,35)(18,30)(19,37,34,38)(20,39,31,40)(22,25)(24,33);(2,3)(6,7)(9,11,22,40,28,31,25,21,14,23,26,39,13,20,12,27)(10,16,33,37,30,19,24,32,15,29,35,38,18,34,17,36)Polycyclic presentation:
2^10|b^2=f,c^2=f,d^2=fh,e^2=fghij,g^2=i,i^2=j,b^a=bh,c^a=chj,c^b=cj,d^a=dh,d^b=dh,e^a=eh,e^b=ehij,e^c=ei,e^d=eg,f^e=fj,g^b=gj,g^c=gj,g^d=gi,g^e=gj,i^d=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72314371, #72314441, #72314574, … |
| maximal normal subgroup of | #72314371, #72314441, #72314574, … |
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.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);