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 | #499831 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5856 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)(3,5)(4,6)(7,8)(16,29)(17,31)(26,32)(27,28);(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16)(13,17)(20,32)(22,27)(23,29)(24,30,36,39)(25,31)(26,33)(28,35)(34,37,40,38);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,26)(17,28)(18,19)(23,33)(24,34)(25,35)(27,31)(29,32)(30,38)(36,40)(37,39);(2,6)(5,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,30)(18,29)(19,32)(21,33)(22,34)(25,36)(27,37)(28,38)(31,39)(35,40);(2,6)(3,7)(10,17)(11,21)(13,25)(15,27)(16,30)(18,31)(19,28)(20,33)(24,36)(26,37)(29,39)(32,38)Polycyclic presentation:
2^10|b^2=fj,c^2=g,h^2=j,b^a=bf,c^b=cj,d^a=dgi,d^b=dgij,e^a=egi,e^b=eghj,e^c=egj,f^d=fj,f^e=fi,h^a=hi,h^b=hij,h^e=hj,h^f=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72318311, #72318432, #72318560, … |
| maximal normal subgroup of | #72318311, #72318432, #72318560, … |
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, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*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, 5, F.8*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);