Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13654 |
| commutator | C32 | C2×C42 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | C22×D32 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 381 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 — 3 generators on 68 points:
(1,2)(3,4)(6,13,17,36)(7,20)(8,26)(9,30)(10,32)(11,28)(14,40)(15,33,47,39)(16,41,51,61)(18,53,52,60)(19,54,50,62)(21,45)(22,29)(23,43)(24,44)(25,42)(27,59)(31,58)(34,63,46,37)(35,64,66,48)(38,67,65,49)(55,56)(57,68);(2,4)(5,6,14,41,9,18,43,65,32,49,57,60,31,51,21,36,11,15,45,63,26,48,68,62,27,50,23,35,22,34,40,33,12,17,42,61,29,52,24,38,59,67,56,53,8,16,7,13,28,47,20,37,58,64,55,54,10,19,44,66,30,46,25,39);(6,15,17,47)(7,21)(13,33,36,39)(14,42)(16,46,51,34)(18,48,52,64)(19,49,50,67)(20,45)(23,44)(24,43)(25,40)(35,60,66,53)(37,61,63,41)(38,62,65,54)(55,68)(56,57)Polycyclic presentation:
2^10|a^2=f,b^2=dehij,c^2=f,d^2=ghj,e^2=hj,g^2=hi,h^2=ij,i^2=j,b^a=bd,c^b=cij,d^a=dg,d^c=dj,e^a=ehij,f^b=fj,g^a=gh,h^a=hi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72286525, #72286777, #72286884, … |
| maximal normal subgroup of | #72286525, #72286777, #72286884, … |
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.4*F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
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.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);