Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C216 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C25 |
| commutator | C24 | C24×C4 |
| prime radical | C28 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
413477 conjugacy classes of subgroups; only the 5957 normal subgroups are shown.
This group has 413477 conjugacy classes of subgroups — too many to draw. Only its 5957 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,8)(17,18,20,19);(1,2,3,4)(5,6,7,8)(10,12)(13,14,15,16)(17,19,20,18);(9,10)(11,12);(13,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^10|a^2=f,b^2=fgj,b^a=bg,c^b=ci,d^b=dh,e^b=egij304 conjugacy classes — view the full character table.
| maximal subgroup of | #72296156, #72296169, #72296173, … |
| maximal normal subgroup of | #72296156, #72296169, #72296173, … |
| maximal quotient of | C2×#72296138, #72335100 |
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, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);