Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #150158 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | #1397 | C23 |
| prime radical | #13697 | C4 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2,6,15,46,51,40,117,54,66,5,36,39,55,116,109,53,77,118,38,68,33,112,70,52,17,76,44,49,16,75,73,11,20,47,71,48,34,106,110,9,18,43,72,45,35,107,111,10,19,41,74,13,64,67,31,8,50,42,14,7,4,30,32)(3,21,24,78,96,87,115,126,104,62,37,113,95,88,123,124,103,63,93,90,57,119,120,108,83,79,94,69,101,22,82,85,29,12,56,59,100,102,92,125,27,80,23,89,91,105,128,127,28,81,99,114,65,86,122,84,26,61,98,58,25,60,97,121);(1,3,11,29)(2,12,20,21)(4,22,16,60)(5,37,43,23)(6,24,47,56)(7,25,49,101)(8,26,52,83)(9,27,54,104)(10,28,53,103)(13,65,68,57)(14,69,44,58)(15,59,71,78)(17,61,50,79)(18,62,66,80)(19,63,77,81)(30,97,75,82)(31,108,70,84)(32,85,73,121)(33,86,64,119)(34,87,51,102)(35,88,55,105)(36,89,72,113)(38,114,74,90)(39,95,45,91)(40,115,106,92)(41,99,118,93)(42,98,76,94)(46,96,48,100)(67,122,112,120)(107,128,116,123)(109,127,111,124)(110,126,117,125);(1,4,8,51,10,55,9,33,11,16,52,34,53,35,54,64)(2,13,66,45,77,48,17,49,20,68,18,39,19,46,50,7)(3,22,83,87,28,105,104,119,29,60,26,102,103,88,27,86)(5,38,47,14,42,71,118,36,43,74,6,44,76,15,41,72)(12,57,62,91,81,96,79,101,21,65,80,95,63,100,61,25)(23,90,56,69,94,59,93,89,37,114,24,58,98,78,99,113)(30,73,112,117,107,111,40,70,75,32,67,110,116,109,106,31)(82,85,120,125,123,127,115,84,97,121,122,126,128,124,92,108)Polycyclic presentation:
2^10|a^2=e,b^2=j,c^2=defi,d^2=fhij,e^2=fgij,f^2=hij,g^2=i,h^2=ij,i^2=j,b^a=bj,c^a=cd,c^b=cgh,d^a=dij,d^b=dj,d^c=df,e^c=ef,e^d=ej,f^a=fj,f^c=fh,g^a=gj,g^c=gi,h^c=hi,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72322819 |
| maximal normal subgroup of | #72322819 |
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.10);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.9);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);