Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #12836 |
| commutator | C22×C16 | C22×C4 |
| Frattini, cosocle | C2×C4×C16 | C23 |
| socle | C2 | #134695 |
| prime radical | C2×#41 | #107 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ C4×#504 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 119 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)(3,11)(4,12,26,53)(5,18)(6,13)(7,14)(8,15)(9,16)(10,17)(19,42,65,99)(20,47)(21,43)(22,44)(23,45)(24,46)(25,78,79,48)(27,49,40,63)(28,50,82,113)(29,51,41,64)(30,52,83,114)(31,62)(32,54)(33,55)(34,56)(35,57)(36,58,87,121)(37,59)(38,60)(39,61)(66,95,88,115)(67,96,94,109)(68,97,77,107)(69,98,80,125)(70,106)(71,100)(72,101)(73,102)(74,103,84,108)(75,104)(76,105)(81,126,122,110)(85,111,92,119)(86,112,93,120)(89,116)(90,117)(91,118)(123,127)(124,128);(1,3,7,22,35,72,91,123,10,24,39,76,9,21,38,71,5,20,31,70,34,73,90,124,8,23,37,75,6,32,33,89)(2,11,14,44,57,101,118,127,17,46,61,105,16,43,60,100,18,47,62,106,56,102,117,128,15,45,59,104,13,54,55,116)(4,25,28,80,27,84,85,19,29,67,86,88,30,81,36,68,26,79,82,69,40,74,92,65,41,94,93,66,83,122,87,77)(12,48,50,98,49,103,111,99,51,109,112,95,52,110,58,107,53,78,113,125,63,108,119,42,64,96,120,115,114,126,121,97);(1,4,6,27,8,29,34,30,5,26,9,40,10,41,35,83)(2,12,13,49,15,51,56,52,18,53,16,63,17,64,57,114)(3,19,21,66,23,68,72,80,20,65,32,88,24,77,73,69)(7,36,33,82,37,92,90,93,31,87,38,28,39,85,91,86)(11,42,43,95,45,97,101,125,47,99,54,115,46,107,102,98)(14,58,55,113,59,119,117,120,62,121,60,50,61,111,118,112)(22,74,71,67,75,122,123,79,70,84,89,94,76,81,124,25)(44,103,100,96,104,110,127,48,106,108,116,109,105,126,128,78)Polycyclic presentation:
2^10|a^2=gi,b^2=f,c^2=e,e^2=g,f^2=hij,g^2=j,h^2=ij,i^2=j,b^a=bgi,c^a=cd,c^b=cdfi,d^a=dj,e^b=ej,f^c=fh,h^c=hi,i^c=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72320551, #72320642, #72320974 |
| maximal normal subgroup of | #72320551, #72320642, #72320974 |
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.7*F.9);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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);