Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #150126 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | #1397 | C23 |
| prime radical | #13697 | C4 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ C4×M64 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,17)(4,24)(5,15)(6,36)(7,40)(8,42)(9,16)(10,19)(11,29)(12,34)(13,27)(14,41)(18,68)(20,79)(21,81)(22,67)(23,66)(25,84)(26,92)(28,98)(30,101)(31,85)(32,62)(33,104)(35,61)(37,109)(38,56)(39,108)(43,64)(44,69)(45,114)(46,117)(47,78)(48,113)(49,76)(50,89)(51,111)(52,96)(53,123)(54,103)(55,72)(57,95)(58,105)(59,88)(60,77)(63,75)(65,73)(70,128)(71,124)(74,116)(80,115)(82,122)(83,125)(86,90)(87,127)(91,126)(93,119)(94,118)(97,107)(99,121)(100,112)(102,106)(110,120);(3,18)(4,25)(10,44)(11,50)(20,70)(21,71)(22,23)(24,84)(28,32)(29,89)(30,90)(31,91)(37,73)(43,80)(45,48)(46,106)(47,49)(52,55)(53,112)(54,121)(57,63)(60,116)(62,98)(72,96)(82,83)(85,126)(86,101)(92,127)(99,103)(100,123)(110,111)(118,119);(1,3,5,20,7,22,34,80,9,18,35,70,41,23,8,43)(2,10,12,45,14,47,15,60,16,44,42,48,40,49,61,116)(4,26,28,93,30,75,31,102,25,87,32,94,90,95,91,117)(6,37,33,83,38,110,88,21,39,73,105,82,97,111,27,71)(11,51,52,122,53,65,121,81,50,120,55,125,112,109,54,124)(13,57,56,118,58,127,36,106,59,63,107,119,104,92,108,46)(17,62,64,24,66,85,128,86,68,98,115,84,67,126,79,101)(19,72,74,29,76,99,113,100,69,96,77,89,78,103,114,123)Polycyclic presentation:
2^10|c^2=e,d^2=fh,e^2=g,f^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bgij,c^a=cd,c^b=cj,d^a=df,d^b=dj,d^c=dfi,e^a=ei,e^d=ej,f^a=fh,f^c=fhj,g^a=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72321873, #72321964, #72321969 |
| maximal normal subgroup of | #72321873, #72321964, #72321969 |
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, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
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*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);