Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #93757 |
| commutator | C32 | C22×C16 |
| Frattini, cosocle | #7330 | C23 |
| socle | C2 | C2×#93757 |
| prime radical | C2×#2262 | C8 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#1311 ◁ C2×#7330 ◁ G | |
| LECS | G ▷ #7330 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
438 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)(3,15,16,55,18,56,62,58,20,54,64,76,37,59,33,107)(4,21,23,78,24,77,85,80)(5,27,29,95)(6,32)(7,36)(8,10)(9,17,40,66,41,67,111,61,39,69,104,90,52,70,51,120)(11,25,47,91,43,88,108,86)(12,30,44,99)(13,34)(14,38)(19,72,74,57,71,46,28,97,75,118,63,123,101,115,98,125)(22,49,81,110,82,112,127,94,83,117,128,42,84,114,68,113)(26,93,87,79,89,53,121,122)(31,103,100,96)(35,105)(45,60)(48,119,106,92,116,65,124,126)(50,73,109,102);(1,3)(2,9)(4,22)(5,28)(6,33)(7,37)(8,20)(10,39)(11,46)(12,49)(13,51)(14,52)(15,53)(16,60)(17,65)(18,38)(19,73)(21,76)(23,82)(24,83)(25,90)(26,81)(27,94)(29,98)(30,101)(31,74)(32,104)(34,64)(35,62)(36,41)(40,105)(42,103)(43,115)(44,117)(45,111)(47,118)(48,97)(50,114)(54,93)(55,77)(56,79)(57,124)(58,78)(59,122)(61,91)(63,100)(66,88)(67,92)(68,121)(69,119)(70,126)(71,99)(72,108)(75,102)(80,107)(84,85)(86,120)(87,127)(89,128)(95,113)(96,110)(106,123)(109,112)(116,125);(2,10)(9,39)(11,43)(12,44)(13,45)(14,36)(15,54)(21,77)(27,95)(32,105)(40,104)(41,52)(42,110)(46,115)(47,108)(48,116)(49,117)(50,109)(51,111)(53,93)(55,76)(56,59)(57,123)(58,107)(72,118)(78,80)(79,122)(94,113)(96,103)(97,125)(106,124)(112,114)Polycyclic presentation:
2^11|a^2=d,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=be,c^a=ck,d^b=dijk,e^a=eg,e^b=egijk,e^d=ej,f^b=fjk,f^e=fk,g^a=gi,g^b=gijk,g^d=gk,h^b=hk,i^a=ij,i^b=ijk,j^a=jk,j^b=jk140 conjugacy classes — view the full character table.
| maximal subgroup of | #72323158, #72323163, #72323166, … |
| maximal normal subgroup of | #72323158, #72323163, #72323166, … |
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.4);;
gap> SetPower(coll, 4, 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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);