Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8534 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×#123 | C23 |
| prime radical | C22×#120 | D4 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
2418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2418 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6,8,5)(4,7)(9,10,13,18,15,20,12,24)(11,16,21,31,23,33,25,38)(14,19,27,35,29,36,26,17)(22,32,37,30,28,40,39,34);(3,8)(5,6)(9,11)(10,16)(12,25)(13,21)(14,28)(15,23)(17,34)(18,31)(19,32)(20,33)(22,29)(24,38)(26,37)(27,39)(30,35)(36,40);(2,5)(3,8)(6,7)(9,12,15,13)(10,17,20,35)(11,21,23,25)(14,26,29,27)(16,30,33,34)(18,19,24,36)(22,37,28,39)(31,40,38,32)Polycyclic presentation:
2^10|a^2=d,c^2=j,d^2=j,e^2=i,b^a=bf,c^a=ce,c^b=chj,d^c=dh,e^a=ehi,e^b=eg,e^c=ei,e^d=ei,f^c=fgij,f^e=fj,g^a=gj,h^a=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72304160 |
| maximal normal subgroup of | #72304160 |
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.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);