Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1450 |
| commutator | C2×C8 | C2×C4×C8 |
| Frattini, cosocle | #1397 | C23 |
| socle | C2 | #149513 |
| prime radical | C23 | #1398 |
| LCS | G ▷ C2×C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #7330 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,4,11,5,13,24,44,6,10,25,37,27,39,71,88,7,12,22,46,28,47,66,99,29,38,67,42,74,90,115,95,8,14,23,45,26,50,68,98,30,40,69,89,72,49,101,102,31,48,65,43,73,92,70,96,75,91,114,87,117,100,116,97)(3,15,17,41,18,52,57,94,19,32,58,76,60,78,110,118,20,51,59,80,61,103,109,121,62,77,111,79,112,119,128,120,16,9,21,33,53,35,63,81,54,36,64,82,104,86,113,123,55,34,56,83,105,84,107,124,106,85,108,93,126,125,127,122);(1,3,8,16)(2,9,14,15)(4,21,23,17)(5,18,26,53)(6,19,30,54)(7,20,31,55)(10,36,40,32)(11,41,45,33)(12,34,48,51)(13,35,50,52)(22,56,65,59)(24,63,68,57)(25,64,69,58)(27,60,72,104)(28,61,73,105)(29,62,75,106)(37,76,89,82)(38,85,91,77)(39,86,49,78)(42,79,87,93)(43,83,46,80)(44,94,98,81)(47,84,92,103)(66,107,70,109)(67,108,114,111)(71,113,101,110)(74,112,117,126)(88,118,102,123)(90,125,100,119)(95,120,97,122)(96,124,99,121)(115,127,116,128);(2,10,12,38,14,40,48,91)(3,16)(4,22,23,65)(5,26)(9,32,34,77,15,36,51,85)(11,42,43,37,45,87,46,89)(13,49,47,100,50,39,92,90)(17,56,21,59)(19,54)(20,55)(24,70,68,66)(25,67,69,114)(27,72)(28,73)(33,79,80,76,41,93,83,82)(35,86,84,125,52,78,103,119)(44,97,96,102,98,95,99,88)(57,109,63,107)(58,108,64,111)(62,106)(71,116,101,115)(74,117)(81,122,121,123,94,120,124,118)(110,128,113,127)Polycyclic presentation:
2^10|a^2=d,b^2=j,c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bg,c^a=chij,c^b=cj,d^b=dj,d^c=di,e^a=eij,e^d=ej,f^c=fj,g^a=gj136 conjugacy classes — view the full character table.
| maximal subgroup of | #72321748, #72321969, #72321979, … |
| maximal normal subgroup of | #72321748, #72321969, #72321979, … |
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.10);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);