Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13654 |
| commutator | C32 | C22×C8 |
| Frattini, cosocle | C4×C32 | C23 |
| socle | C2 | #151080 |
| prime radical | #13741 | C4 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C16 ◁ C8×C16 ◁ #7329 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,15,8,17,45,18,11,19,9,49,10,50,46,69)(3,20,24,72,26,74,92,128,28,76,44,110,27,98,93,75)(4,16,31,60,32,61,7,47,35,68,33,101,34,13,48,59)(5,36,38,103,40,62,107,104,43,105,41,109,42,52,108,116)(12,51,54,90,55,114,121,23,57,87,67,89,112,115,56,91)(14,37,63,21,64,78,65,79,66,30,100,81,113,80,102,39)(22,82,25,94,29,73,95,127,85,124,97,126,84,70,96,125)(53,117,118,83,119,88,71,99,120,77,123,86,58,122,111,106);(1,3,21,77,7,25,41,87,45,93,30,99,34,22,5,23,10,27,39,88,31,96,107,114,6,44,78,83,35,84,42,90,11,28,81,117,48,97,38,51,46,92,37,106,32,85,43,91,8,26,79,122,33,95,108,115,9,24,80,86,4,29,40,89)(2,12,52,82,16,53,65,75,49,54,104,94,101,118,14,20,17,55,36,73,61,119,100,72,69,121,109,127,59,71,64,74,19,57,62,124,68,120,102,128,15,67,116,126,60,123,66,76,50,112,105,70,13,58,63,110,18,56,103,125,47,111,113,98);(1,4,9,33,8,32,46,48,11,35,6,31,10,34,45,7)(2,13,15,59,17,16,18,60,19,61,49,47,50,68,69,101)(3,22,44,97,26,29,93,96,28,85,24,25,27,84,92,95)(5,37,41,81,40,78,108,39,43,30,38,21,42,80,107,79)(12,53,54,118,55,119,121,71,57,120,67,123,112,58,56,111)(14,62,63,104,64,105,65,109,66,52,100,116,113,36,102,103)(20,70,72,125,74,82,128,94,76,73,110,127,98,124,75,126)(23,86,51,88,89,106,114,77,91,83,87,122,90,99,115,117)Polycyclic presentation:
2^10|a^2=e,b^2=dfij,c^2=egij,d^2=fhj,e^2=g,f^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bd,c^a=ci,c^b=cij,d^a=df,d^c=dj,e^b=ej,f^a=fh,h^a=hi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72230747, #72230779, #72230794 |
| maximal normal subgroup of | #72230747, #72230779, #72230794 |
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.5);;
gap> SetPower(coll, 2, F.4*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*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);