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 |
| prime radical | #6442565 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15,21,27)(12,16)(13,17)(14,18)(19,31,33,25)(20,26,34,37)(22,28)(23,29)(24,30)(32,39,40,36)(35,38);(11,19)(13,24)(15,25)(17,30)(20,32)(21,33)(23,35)(26,36)(27,31)(29,38)(34,40)(37,39);(2,5)(6,7)(9,11)(10,15,18,27)(12,20)(13,19,24,33)(14,21)(16,26,28,37)(17,25)(22,34)(23,32,35,40)(29,36)(30,31)(38,39);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,20)(13,23)(14,22)(15,26)(17,29)(18,28)(19,32)(21,34)(24,35)(25,36)(27,37)(30,38)(31,39)(33,40);(12,22)(16,28)(20,34)(23,35)(26,37)(29,38)(32,40)(36,39)Polycyclic presentation:
2^10|a^2=f,c^2=h,d^2=fi,g^2=j,b^a=bi,c^a=cfh,c^b=cg,d^a=dfi,e^d=ej,f^c=fj,g^a=gi,g^b=gj,g^c=gij,h^a=hj,h^b=hi,h^g=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72318548, #72318828, #72318927, … |
| maximal normal subgroup of | #72318548, #72318828, #72318927, … |
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.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);