Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152629 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | #512945 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,6,14,41,30,95,100,9,15,43,34,59,49,38,27,46,57,5,13,36,64,109,97,111,101,39,40,44,32,94,93,7,17,42,66,35,63,108,117,45,67,110,106,12,31,58,98,10,16,37,65,60,48,107,112,47,29,62,99,8,4,26,28)(3,18,21,68,85,115,128,124,24,70,87,79,51,78,104,74,73,55,33,52,103,91,125,122,121,76,105,113,88,19,72,75,22,11,50,53,80,77,120,123,89,54,114,102,56,92,82,96,25,71,20,61,81,116,126,127,90,119,83,84,23,69,86,118);(1,3)(2,11)(4,19)(5,33)(6,21)(7,22)(8,23)(9,24)(10,25)(12,56)(13,61)(14,53)(15,54)(16,55)(17,18)(20,37)(26,86)(27,96)(28,75)(29,76)(30,77)(31,78)(32,69)(34,102)(35,80)(36,103)(38,104)(39,105)(40,84)(41,85)(42,50)(43,87)(44,88)(45,89)(46,73)(47,90)(48,91)(49,92)(51,59)(52,65)(57,71)(58,82)(60,81)(62,83)(63,115)(64,116)(66,68)(67,70)(72,94)(74,98)(79,106)(93,118)(95,128)(97,127)(99,113)(100,123)(101,119)(107,126)(108,120)(109,125)(110,114)(111,121)(112,122)(117,124);(1,4,10,49,7,32,46,31)(2,12,57,44,17,59,16,8)(3,19,73,78,22,69,25,92)(5,34,6,40,37,106,42,99)(9,48,47,30,45,64,111,63)(11,51,55,88,18,56,71,23)(13,62,66,110,65,39,14,43)(15,35,101,36,67,41,29,60)(20,79,21,84,33,102,50,113)(24,91,121,115,89,116,90,77)(26,93,58,98,94,28,38,27)(52,83,68,87,61,105,53,114)(54,85,76,103,70,80,119,81)(72,75,82,74,86,118,104,96)(95,97,107,117,108,112,109,100)(120,122,126,124,128,127,125,123)Polycyclic presentation:
2^10|a^2=e,c^2=defg,d^2=ghj,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bj,c^a=cd,c^b=cf,d^b=dj,d^c=dg,e^c=egj,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij148 conjugacy classes — view the full character table.
| maximal subgroup of | #72322773, #72322943, #72322977 |
| maximal normal subgroup of | #72322773, #72322943, #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.5);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8);;
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.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
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);