Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #322 |
| commutator | C8 | C8×C16 |
| Frattini, cosocle | C2×#1397 | C22 |
| socle | C22 | #7332 |
| prime radical | C23 | #1398 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ C2×#1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(5,6,8,17,11,19,33,43,13,20,35,39,41,60,54,65,14,21,36,29,46,61,32,26,9,38,31,50,16,51,53,66,15,22,37,56,48,30,63,55,12,49,34,25,47,64,68,52,40,58,67,7,10,24,44,27,42,18,57,28,45,59,23,62);(2,4)(5,7,23,6,16,52,31,59,42,25,44,18,46,55,36,24,40,56,68,58,41,66,35,64,12,50,63,49,11,26,8,30,15,29,53,22,45,65,57,51,9,39,32,38,10,43,67,61,14,17,54,21,47,62,34,60,13,28,33,20,48,27,37,19)Polycyclic presentation:
2^10|a^2=c,b^2=cdei,c^2=f,d^2=ij,e^2=g,f^2=h,g^2=i,h^2=i,i^2=j,b^a=bd,c^b=cij,d^a=dj,d^b=dj,e^a=eij,e^b=eij,e^c=ej,f^b=fj,g^a=gj,g^b=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72286426, #72323180, #72323181 |
| maximal normal subgroup of | #72286426, #72323180, #72323181 |
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.5*F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, 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);