Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C25 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5962 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)(3,7)(4,6)(5,8)(12,24)(17,18)(21,34)(25,37)(28,31)(29,32)(35,40)(38,39);(1,2)(3,7)(4,6)(5,8)(9,10,14,19)(11,20,22,15)(12,16)(13,18)(17,26)(21,33)(23,31)(24,30)(25,29,37,32)(27,34)(28,36)(35,38,40,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,23)(14,22)(16,27)(17,31)(18,28)(19,20)(24,34)(25,35)(26,36)(29,39)(30,33)(32,38)(37,40);(2,6)(5,7)(9,12)(10,16)(11,21)(13,25)(14,24)(15,27)(17,32)(18,29)(19,30)(20,33)(22,34)(23,35)(26,37)(28,39)(31,38)(36,40);(3,8)(5,7)(10,17,19,18)(11,22)(12,24)(15,28,20,31)(16,29,30,32)(23,36)(25,37)(27,38,33,39)Polycyclic presentation:
2^10|b^2=fj,c^2=g,e^2=i,h^2=j,b^a=bf,c^a=cg,c^b=cgj,d^a=dgij,d^b=dgij,e^a=ei,e^b=eh,e^c=egj,e^d=ej,f^d=fj,f^e=fi,h^a=hi,h^b=hij,h^f=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72318267, #72318445, #72318462, … |
| maximal normal subgroup of | #72318267, #72318445, #72318462, … |
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*F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);