Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #1455 |
| commutator | C22×C8 | C2×C4×C8 |
| Frattini, cosocle | C2×#1397 | C23 |
| socle | C2 | #72322700 |
| prime radical | C24 | #1398 |
| LCS | G ▷ C22×C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ C2×#1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,4,10,5,12,23,41,6,14,24,43,27,50,72,100,7,11,25,44,28,45,73,101,30,47,74,103,79,105,121,128,8,13,22,42,29,49,68,99,31,51,69,102,80,110,118,127,32,46,70,40,81,92,114,96,82,106,83,97,123,125,124,126)(3,15,17,53,18,55,59,112,19,34,60,86,61,48,107,108,20,54,21,64,62,109,66,116,63,87,67,88,115,104,117,98,16,52,56,111,26,75,76,122,57,9,33,35,77,37,84,89,58,38,65,90,78,93,71,119,113,36,85,39,94,91,120,95);(1,3)(2,9,11,36,13,34,46,87)(4,21,22,65)(5,26)(6,19)(7,20)(8,16)(10,39,40,35,42,88,44,86)(12,48,45,104,49,37,92,91)(14,38,47,15,51,54,106,52)(17,25,56,70)(18,29)(23,71,68,66)(24,67,69,85)(27,77)(28,78)(30,63)(31,57)(32,58)(33,83,60,74)(41,98,96,108,99,95,101,89)(43,53,97,90,102,111,103,64)(50,109,105,75,110,93,125,55)(59,114,76,73)(61,80)(62,81)(72,120,118,117)(79,94)(82,113)(84,121,107,124)(100,122,126,116,127,112,128,119)(115,123);(9,34)(15,52)(35,86)(36,87)(37,48)(38,54)(39,88)(53,111)(55,75)(64,90)(89,108)(91,104)(93,109)(95,98)(112,122)(116,119)Polycyclic presentation:
2^11|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bij,c^a=cd,c^b=ch,d^b=dk,e^b=ej,f^a=fjk,f^e=fk,g^b=gk,h^a=hk224 conjugacy classes — view the full character table.
| maximal subgroup of | #72322743, #72323081, #72323128, … |
| maximal normal subgroup of | #72322743, #72323081, #72323128, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 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, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);