Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C32 · Schur multiplier (H2) = C82 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | C82 |
| commutator | C8 | C2×C8×C32 |
| Frattini, cosocle | C4×#1397 | C23 |
| socle | C22 | #72234237 |
| prime radical | C23 | #151879 |
| LCS | G ▷ C8 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×#1397 ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 160 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 160 points:
(1,2)(3,11,13,24)(4,8)(5,10)(6,9)(7,14,19,27)(12,23,25,31)(15,21)(16,20)(17,22)(18,26,29,32)(28,30)(33,34,37,46,40,49,70,96,39,48,69,95,65,91,126,148,42,50,72,97,79,104,124,154,77,102,68,131,66,93,127,150,43,52,73,99,80,90,133,151,76,47,100,101,64,92,125,149,41,51,71,98,63,105,128,155,38,74,75,134,36,45,62,89)(35,53,55,106,58,107,117,147,57,86,116,142,112,145,123,153,60,108,118,156,111,82,132,139,121,88,115,143,54,103,110,157,61,84,119,144,78,136,137,160,120,87,135,94,113,146,158,152,59,109,67,129,122,140,130,159,56,44,81,83,114,85,138,141);(1,3)(2,7,9,19)(4,12)(5,14)(6,13)(8,18,20,29)(10,11,22,24)(15,26)(16,25)(17,27)(21,23,30,31)(28,32)(33,35,38,56,41,59,76,120,43,61,77,121,42,60,39,57)(34,44,47,84,50,86,74,109,52,88,48,53,51,87,102,108)(36,54,63,111,64,112,80,58,66,114,79,122,65,113,40,78)(37,67,68,116,71,119,69,81,73,118,75,135,72,55,100,115)(45,82,90,85,91,136,105,145,93,140,49,103,92,107,104,146)(46,94,95,129,97,83,131,106,99,142,101,156,98,143,134,144)(62,123,124,137,125,138,70,132,127,158,128,117,126,110,133,130)(89,147,96,153,148,139,154,157,150,160,151,152,149,159,155,141);(1,4,5,15,6,16,17,28)(2,8,10,21,9,20,22,30)(3,12,14,26,13,25,27,32)(7,18,11,23,19,29,24,31)(33,36,38,63,41,64,76,80,43,66,77,79,42,65,39,40)(34,45,74,105,51,92,47,90,52,93,102,104,50,91,48,49)(35,54,56,111,59,112,120,58,61,114,121,122,60,113,57,78)(37,62,75,128,71,125,100,133,73,127,68,124,72,126,69,70)(44,82,109,145,87,107,84,85,88,140,108,146,86,136,53,103)(46,89,134,155,98,149,101,151,99,150,131,154,97,148,95,96)(55,110,81,132,67,123,135,117,119,138,115,130,118,158,116,137)(83,139,129,153,94,147,144,141,143,159,156,152,142,160,106,157)Polycyclic presentation:
2^12|a^2=d,b^2=e,c^2=fkl,d^2=g,e^2=h,f^2=il,g^2=j,h^2=l,i^2=l,j^2=k,k^2=l,b^a=bf,d^b=di,e^a=eil,e^d=el,g^b=gl,h^a=hl736 conjugacy classes — view the full character table.
| maximal subgroup of | #72234405, #72234407 |
| maximal normal subgroup of | #72234405, #72234407 |
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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.11*F.12);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.12);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);