Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #11680 |
| commutator | C22×C16 | C22×C4 |
| Frattini, cosocle | C2×#346 | C23 |
| socle | C2 | #129555 |
| prime radical | #7623243 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C4×C8 ◁ C22×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11,6,13,24,44,32,43,31,62,30,14,5,28,9,16,27,46,33,49,59,12,8,15,26,45,7,34,25,60)(3,17,19,41,21,47,55,48,35,63,61,10,36,37,53,39,18,52,29,51,54,64,57,40,23,42,56,38,22,58,20,50);(1,3)(2,10)(4,20)(5,29)(6,22)(7,21)(8,35)(9,18)(11,42)(12,47)(13,48)(14,50)(15,51)(16,38)(17,45)(19,25)(23,32)(24,56)(26,55)(27,53)(28,37)(30,54)(31,57)(33,36)(34,39)(40,49)(41,43)(44,64)(46,63)(52,62)(58,60)(59,61);(3,18)(19,29)(20,53)(21,54)(22,36)(23,35)(55,57)(56,61)Polycyclic presentation:
2^10|a^2=d,d^2=g,f^2=hj,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=ce,c^b=cghj,d^b=dh,e^b=ej,f^a=fi,f^b=fhi,f^d=fj,g^b=gi,h^a=hj,h^b=hij,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72225045, #72225066, #72225074, … |
| maximal normal subgroup of | #72225045, #72225066, #72225074, … |
| maximal quotient of | C2×#72225035 |
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, 4, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
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.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);