Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #322 |
| commutator | C8 | C8×C16 |
| Frattini, cosocle | C2×#1397 | C22 |
| socle | C2 | #93990 |
| prime radical | C23 | #1398 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ C2×#1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 2 generators on 128 points:
(1,2,5,12,6,14,32,54,7,16,33,56,36,62,88,113,8,17,34,53,37,63,89,110,29,51,79,105,81,108,122,127,9,15,31,55,38,61,85,112,39,49,86,106,94,107,124,128,26,13,57,58,78,59,114,115,4,11,27,48,28,50,80,104)(3,18,20,64,21,65,72,119,22,44,73,97,74,60,116,117,23,66,30,82,75,118,84,123,19,42,68,99,69,100,120,111,24,47,71,103,35,90,91,126,70,10,40,41,92,43,95,96,67,45,83,98,93,101,87,125,25,46,76,52,77,102,121,109);(1,3)(2,10,13,42,15,44,17,46)(4,25)(5,30,31,83)(6,35)(7,22)(8,23)(9,24)(11,47,49,66,51,18,16,45)(12,52,53,97,55,99,58,41)(14,60,59,102,61,43,63,100)(19,29)(20,34,71,57)(21,38)(26,67)(27,73,79,40)(28,69)(32,87,85,84)(33,68,86,76)(36,92)(37,93)(39,70)(48,98,56,64,105,82,106,103)(50,65,107,101,108,90,62,118)(54,111,110,96,112,109,115,117)(72,114,91,89)(74,94)(75,78)(77,81)(80,95,122,116)(88,121,124,120)(104,123,113,126,127,125,128,119)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=ij,d^2=f,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bc,c^b=cj,d^b=di,e^a=ei,e^d=ej,f^b=fj,g^a=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72321587, #72321671, #72321676, … |
| maximal normal subgroup of | #72321587, #72321671, #72321676, … |
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.9*F.10);;
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.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
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);