Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152435 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | C2×#2262 | C22 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,5,11,29,45,86,105,8,13,32,47,40,52,80,85,36,4,23,25,74,81,107,124,94,27,76,82,7,12,31,46,6,15,30,49,39,51,91,92,35,54,89,108,95,26,75,28,9,14,33,48,87,37,96,84,38,50,90,106,34,53,88,93)(3,16,19,55,66,103,118,127,21,57,68,110,73,78,114,83,64,17,58,59,111,97,128,126,104,61,113,115,20,56,67,109,18,10,41,42,72,100,121,122,63,44,99,102,116,60,112,65,22,24,69,77,119,70,123,117,71,79,120,125,62,43,98,101);(1,3,6,18)(2,10)(4,24)(5,19,30,41)(7,20,34,62)(8,21,35,63)(9,22,36,64)(11,42)(12,43)(13,44)(14,17)(15,16)(23,58,33,69)(25,77)(26,78)(27,79)(28,83)(29,66,39,72)(31,67,88,98)(32,68,89,99)(37,97)(38,71,94,104)(40,73,95,116)(45,100)(46,101)(47,102)(48,59)(49,55)(50,61)(51,103)(52,60)(53,56)(54,57)(65,85)(70,81)(74,111,87,119)(75,112,80,114)(76,113,90,120)(82,125)(84,126)(86,118,91,121)(92,127)(93,109)(96,123,107,128)(105,122)(106,115)(108,110)(117,124);(1,4,9,15,6,14,36,2)(3,17,22,10,18,24,64,16)(5,28,33,93,30,85,23,46)(7,37,40,45,34,81,95,51)(8,13,38,27,35,54,94,50)(11,31,25,80,49,88,48,75)(12,39,26,74,53,29,52,87)(19,65,69,101,41,83,58,109)(20,70,73,103,62,97,116,100)(21,57,71,61,63,44,104,79)(32,92,90,84,89,105,76,124)(42,98,77,112,55,67,59,114)(43,66,78,119,56,72,60,111)(47,107,82,86,108,96,106,91)(68,122,120,117,99,127,113,126)(102,123,125,121,110,128,115,118)Polycyclic presentation:
2^10|a^2=e,b^2=f,c^2=fij,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bd,c^a=ceij,c^b=cd,d^b=dj,e^c=eg,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72321641, #72322926, #72322977 |
| maximal normal subgroup of | #72321641, #72322926, #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, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
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.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
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);