Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #153594 |
| commutator | C64 | C22×C4 |
| Frattini, cosocle | C2×C64 | C23 |
| prime radical | #153706 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ #7510 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,3,11,19,44,60,86,5,14,20,46,66,102,72,108,27,54,70,101,77,126,123,111,75,109,116,112,25,52,62,57,7,16,22,43,68,100,117,110,28,55,71,88,79,99,115,36,31,49,65,105,119,106,120,53,76,97,113,34,26,42,58,9,8,12,18,45,63,98,114,35,29,48,64,104,118,95,121,56,81,41,90,93,127,128,124,96,82,94,125,83,78,39,73,32,30,10,37,38,80,91,122,84,74,40,89,92,24,51,61,87,6,15,21,47,67,103,69,107,23,50,59,85,4,13,17,33);(2,9)(3,17)(4,19)(5,23)(6,27)(7,30)(10,32)(11,42)(12,33)(13,45)(14,53)(15,56)(16,57)(18,58)(20,69)(21,72)(22,73)(24,77)(25,80)(26,63)(28,82)(29,76)(31,81)(34,44)(35,50)(36,54)(37,62)(38,39)(40,96)(41,87)(43,52)(46,106)(47,95)(48,107)(49,108)(51,93)(55,111)(59,60)(61,70)(64,120)(65,121)(66,67)(68,78)(71,124)(74,75)(79,127)(83,91)(84,94)(85,98)(86,97)(88,126)(89,123)(90,115)(92,128)(99,101)(100,112)(102,105)(103,104)(109,110)(113,114)(116,122)(117,125)(118,119);(2,10,12,16)(3,18)(9,32,33,57)(11,43,45,38)(13,39,42,52)(14,40,48,55)(15,41,49,54)(17,58)(20,64)(21,65)(22,37)(34,83,85,112)(35,84,86,110)(36,56,87,108)(44,91,98,100)(46,88,104,92)(47,101,105,93)(50,94,97,109)(51,95,99,102)(53,96,107,111)(59,113)(60,114)(61,115)(62,73)(69,120)(70,90)(71,89)(72,121)(103,128,106,126)(116,125)(117,122)(123,124)Polycyclic presentation:
2^10|a^2=d,c^2=e,d^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bdj,c^a=ci,d^b=df,d^c=dj,e^a=ej,f^b=fg,g^b=gh,h^b=hi,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72230779 |
| maximal normal subgroup of | #72230779 |
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, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 6, 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.4*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);