Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #94752 |
| commutator | C64 | C2×C8 |
| Frattini, cosocle | #7510 | C22 |
| prime radical | C22 | #7548 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ #7510 ◁ G | |
| LECS | G ▷ #7510 ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,3,10,18,41,69,60,5,14,20,45,25,56,67,9,29,34,75,91,81,115,74,38,89,92,126,97,4,12,17,43,7,16,22,42,23,55,66,109,30,62,77,36,80,40,78,35,33,59,73,52,85,102,125,94,24,53,65,107,26,57,68,99,8,13,19,44,71,105,122,95,31,58,72,93,87,103,119,37,90,51,112,113,83,114,120,98,84,100,121,128,27,54,64,108,32,11,47,48,82,104,118,117,88,50,111,96,127,101,124,63,6,15,21,46,70,106,123,116,28,61,76,110,86,49,79,39);(2,9,11,35,13,37,16,63)(3,17,19,64)(4,23)(5,28)(6,29)(7,32)(10,40,42,56,44,101,48,103)(12,52,49,113,54,46,57,91)(14,60,50,109,58,95,62,117)(15,43,51,99,59,108,34,39)(18,26)(20,74,72,120)(21,78,73,124)(22,79,47,68)(24,31)(25,85)(27,82)(30,84)(33,90)(36,41,93,104,96,105,45,55)(38,100,94,53,98,92,116,61)(65,118,76,66)(67,112,119,75)(69,121,122,126)(70,87)(71,86)(77,123,111,125)(80,81)(83,127)(88,89)(97,115,107,106,128,114,110,102)Polycyclic presentation:
2^10|a^2=c,b^2=d,c^2=egij,d^2=f,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bch,c^b=ce,d^a=di,d^c=dj,e^b=eg,f^a=fj,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.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
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.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
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);