Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #322 |
| commutator | C8 | C8×C16 |
| Frattini, cosocle | C2×#1311 | C22 |
| socle, prime radical | C23 | #1313 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ C2×#1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 2 generators on 52 points:
(1,2)(3,4)(5,6,8,11,9,12,16,15,10,13,17,19,18,20,14,7)(22,27,29,41,31,38,32,43)(23,33,35,50)(24,37)(28,44,46,36)(30,48,34,45,49,51,47,42)(39,52);(2,4)(5,7,14,20,18,19,17,13,10,15,16,12,9,11,8,6)(21,22,28,45,39,41,33,49,40,29,44,48,24,38,23,34,26,31,46,42,52,43,50,30,25,32,36,51,37,27,35,47)Polycyclic presentation:
2^10|a^2=c,b^2=cdgij,c^2=f,d^2=g,e^2=i,f^2=h,g^2=i,i^2=j,b^a=be,c^b=ci,d^a=di,d^b=di,d^c=dj,e^a=ej,e^b=ej,f^b=fj,g^a=gj,g^b=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72286374, #72287051, #72287134 |
| maximal normal subgroup of | #72286374, #72287051, #72287134 |
| maximal quotient of | #72286387 |
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.3);;
gap> SetPower(coll, 2, F.3*F.4*F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);