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 | #7623621 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ #20493 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3009 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)(3,4)(5,7)(6,8)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,31)(22,32)(23,27)(24,35)(25,34)(26,30)(28,36)(29,33)(37,39)(38,40);(1,3,5,6)(2,4,7,8)(9,11,13,20)(10,16,17,15)(12,23)(14,30,29,22)(18,27)(19,26,33,32)(21,28)(24,35)(25,34)(31,36)(37,39)(38,40);(1,4,5,8)(2,6,7,3)(10,17)(12,24)(14,26)(15,16)(18,34)(19,22)(21,37)(23,35)(25,27)(28,39)(29,32)(30,33)(31,38)(36,40);(1,2)(3,8)(4,6)(5,7)(11,20)(12,25)(14,26)(15,16)(18,34)(19,30)(21,37)(22,33)(23,35)(24,27)(28,38)(29,32)(31,39)(36,40);(2,7)(4,8)(9,12,26,38,15,28,29,25,13,27,32,39,16,31,14,24)(10,18,30,40,20,36,33,34,17,23,22,37,11,21,19,35)Polycyclic presentation:
2^10|b^2=f,c^2=hj,e^2=ghij,g^2=ij,i^2=j,c^a=ch,c^b=chj,d^b=dh,e^a=ehj,e^b=eij,e^c=eg,e^d=egj,f^e=fj,g^b=gj,g^c=gi,g^d=gi,i^c=ij,i^d=ij103 conjugacy classes — view the full character table.
| maximal subgroup of | #72314059, #72314371, #72314408, … |
| maximal normal subgroup of | #72314059, #72314371, #72314408, … |
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.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
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.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);