Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C82 |
| commutator | C8 | C2×C82 |
| Frattini, cosocle | #1397 | C23 |
| socle | C2 | C2×#7330 |
| prime radical | C23 | #1398 |
| LCS | G ▷ C8 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 179 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:
(1,2,5,14,7,16,37,63,9,18,39,62,46,71,89,120,10,17,40,64,4,13,30,56,6,42,43,99,32,59,90,113,11,19,41,65,33,58,91,114,29,15,66,67,31,68,93,121,8,47,38,98,34,60,92,115,44,69,36,96,88,57,97,116)(3,20,22,73,24,74,83,125,26,12,48,50,78,52,106,109,27,54,35,94,81,112,95,127,23,53,86,61,87,108,126,118,28,75,85,117,45,101,102,128,76,55,100,110,103,70,122,123,25,51,84,111,21,72,77,124,79,104,82,119,80,49,105,107);(1,3,6,23,8,25,9,26,11,28,44,79,10,27,29,76)(2,12,15,51,17,53,69,20,19,55,18,54,47,104,42,75)(4,21,31,78,7,45,32,80,34,81,46,103,33,24,88,87)(5,35,36,48,38,22,66,82,41,84,43,100,40,85,39,86)(13,49,57,101,58,52,71,72,60,108,59,74,16,70,68,112)(14,61,62,117,64,110,99,111,65,119,67,73,98,50,96,94)(30,83,89,105,37,77,97,122,92,102,93,126,91,95,90,106)(56,109,113,127,114,118,121,128,115,123,116,124,63,107,120,125);(1,4,29,88,10,33,44,46,11,34,9,32,8,7,6,31)(2,13,15,57,17,58,69,71,19,60,18,59,47,16,42,68)(3,21,76,87,27,24,79,103,28,81,26,80,25,45,23,78)(5,30,66,97,40,91,36,89,41,92,39,90,38,37,43,93)(12,49,51,101,53,52,20,72,55,108,54,74,104,70,75,112)(14,56,67,116,64,114,96,120,65,115,62,113,98,63,99,121)(22,77,100,126,35,83,82,122,85,95,48,105,84,102,86,106)(50,107,111,128,61,109,73,124,110,118,94,125,119,123,117,127)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=hj,d^2=f,e^2=g,f^2=h,g^2=j,h^2=i,i^2=j,b^a=bhij,d^b=di,e^a=eij,e^d=ej,f^b=fj,g^a=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72234403, #72321339, #72321447, … |
| maximal normal subgroup of | #72234403, #72321339, #72321447, … |
| maximal quotient of | #72234385 |
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, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, 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, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);