Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #322 |
| commutator | C8 | C2×C4×C16 |
| Frattini, cosocle | #1397 | C23 |
| socle | C2 | C2×#7332 |
| prime radical | C23 | #1398 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2,4,10,5,12,23,41,6,14,24,43,27,50,73,99,7,11,25,44,28,45,74,100,30,47,75,102,79,103,121,128,8,13,22,42,29,49,70,98,31,51,71,101,80,108,118,127,32,46,59,40,81,89,112,95,82,104,113,96,123,125,124,126)(3,15,17,53,18,55,60,109,19,34,61,84,63,48,105,106,20,54,62,110,64,107,72,119,65,85,114,39,91,92,120,93,16,52,56,94,26,76,77,122,57,9,33,35,78,37,83,86,58,38,21,66,67,90,68,115,111,36,69,87,116,88,117,97);(1,3)(2,9,11,36,13,34,46,85)(4,21,22,62)(5,26)(6,19)(7,20)(8,16)(10,39,40,84,42,87,44,35)(12,48,45,92,49,37,89,88)(14,38,47,15,51,54,104,52)(17,59,56,25)(18,29)(23,72,70,68)(24,69,71,114)(27,78)(28,67)(30,65)(31,57)(32,58)(33,75,61,113)(41,97,95,86,98,93,100,106)(43,94,96,110,101,53,102,66)(50,107,103,76,108,90,125,55)(60,74,77,112)(63,80)(64,81)(73,120,118,117)(79,116)(82,111)(83,124,105,121)(91,123)(99,109,126,115,127,122,128,119);(3,16)(9,34)(15,52)(17,56)(18,26)(19,57)(20,58)(21,62)(33,61)(35,84)(36,85)(37,48)(38,54)(39,87)(53,94)(55,76)(60,77)(63,78)(64,67)(65,111)(66,110)(68,72)(69,114)(83,105)(86,106)(88,92)(90,107)(91,116)(93,97)(109,122)(115,119)(117,120)Polycyclic presentation:
2^10|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bghij,c^b=cj,d^b=dij,e^a=eij,e^d=ej,f^b=fj,g^a=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72321748, #72321894, #72321955, … |
| maximal normal subgroup of | #72321748, #72321894, #72321955, … |
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.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*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);