Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #153322 |
| commutator, Frattini, cosocle | C22×C32 | C23 |
| prime radical | C23 | C2×SD64 |
| LCS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2,5,12,30,50,89,97,9,14,33,52,44,56,81,36,41,4,24,25,77,82,109,96,98,27,80,83,8,13,32,51,7,16,6,35,31,55,90,93,40,58,38,95,91,26,79,29,10,15,34,53,78,42,99,88,43,54,92,108,39,57,37,94)(3,17,19,46,67,101,120,122,22,60,69,105,76,64,112,66,72,49,87,63,116,73,124,119,110,106,126,118,21,59,68,104,20,11,28,45,75,100,121,127,71,48,86,103,123,107,117,84,23,18,61,62,111,114,128,125,74,65,113,115,70,47,85,102);(1,3,7,20)(2,11)(4,18)(5,28,6,19)(8,21,39,70)(9,22,40,71)(10,23,41,72)(12,46)(13,47)(14,48)(15,49)(16,17)(24,61,34,87)(25,63)(26,64)(27,65)(29,84)(30,67,31,75)(32,85,37,68)(33,86,38,69)(35,45)(36,66)(42,73)(43,74,98,110)(44,76,91,123)(50,100)(51,104)(52,105)(53,62)(54,106)(55,101)(56,107)(57,59)(58,60)(77,116,78,111)(79,112,81,117)(80,113,92,126)(82,114)(83,118)(88,125)(89,121,90,120)(93,127)(94,102)(95,103)(96,119)(97,122)(99,124,109,128)(108,115);(1,4,10,16,7,15,41,2)(3,18,23,17,20,49,72,11)(5,29,34,94,6,36,24,51)(8,42,44,50,39,82,91,55)(9,14,43,27,40,58,98,54)(12,32,25,81,35,37,53,79)(13,31,26,77,57,30,56,78)(19,66,61,104,28,84,87,102)(21,73,76,100,70,114,123,101)(22,48,74,65,71,60,110,106)(33,93,92,88,38,97,80,96)(45,68,62,112,46,85,63,117)(47,75,64,116,59,67,107,111)(52,109,83,89,95,99,108,90)(69,122,113,119,86,127,126,125)(103,124,115,120,105,128,118,121)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=fij,d^2=ghj,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bdegj,c^a=cdij,c^b=cdefg,d^b=dj,d^c=dg,e^a=ej,e^c=egj,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72322063, #72322907, #72322977 |
| maximal normal subgroup of | #72322063, #72322907, #72322977 |
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.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.7*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*F.5*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
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);