Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23×C42 |
| commutator | C2×C4 | C23×C4×C8 |
| Frattini, cosocle | C4×C16 | C25 |
| socle | C22 | C23×#346 |
| prime radical | #3444 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,44)(30,45)(31,40)(32,46)(33,47)(34,41)(35,48)(36,49)(37,50)(38,42)(39,43)(51,65)(52,67)(53,68)(54,69)(55,62)(56,70)(57,63)(58,71)(59,64)(60,72)(61,66)(73,78)(74,79)(75,80)(76,77);(2,6)(7,12)(8,13)(14,16);(18,24)(25,40)(26,41)(27,42)(28,43)(44,47)(45,62)(46,63)(48,64)(49,65)(50,66)(67,69)(68,71)(70,77)(72,78)(79,80);(17,19,20,30,21,32,35,56,22,33,36,54,37,58,60,75,23,31,34,55,38,57,59,76,39,29,51,52,61,53,73,74)(18,25,26,45,27,46,48,70,28,47,49,69,50,71,72,80,24,40,41,62,42,63,64,77,43,44,65,67,66,68,78,79);(1,3,4,9,5,10,11,15)(2,7,8,14,6,12,13,16)(19,29,31,33)(20,34)(25,44,40,47)(26,41)(30,54,55,52)(32,53,57,58)(35,59)(36,51)(45,69,62,67)(46,68,63,71)(48,64)(49,65)(56,75,76,74)(60,73)(70,80,77,79)(72,78)Polycyclic presentation:
2^11|d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bi,c^a=ck,e^d=ej,f^e=fk,g^d=gk416 conjugacy classes — view the full character table.
| maximal subgroup of | #72231918, #72234318, #72321025, … |
| maximal normal subgroup of | #72231918, #72234318, #72321025, … |
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, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);