Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
566 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,5)(4,11)(6,10)(7,13)(8,14)(9,15)(12,16)(17,18)(19,30)(20,36,40,50)(21,41)(22,28)(23,29)(24,33)(25,39,51,65)(26,43)(27,42)(31,56,59,45)(32,60)(34,47)(35,54)(37,48,58,73)(38,49,68,74)(44,52)(46,63)(53,70)(55,61)(57,77,79,72)(62,75)(64,80,69,67)(66,76,78,71);(1,3,5,2)(4,7,12,9)(6,10,14,8)(11,15,16,13)(17,19,22,33,23,35,42,34)(18,24,27,30,29,47,28,54)(20,37,65,31,40,58,39,59)(21,32,43,61,44,63,70,62)(25,48,36,45,51,73,50,56)(26,46,41,55,53,60,52,75)(38,66,80,57,68,78,67,79)(49,72,64,71,74,77,69,76);(1,4,6,13,5,12,14,15)(2,7,8,11,3,9,10,16)(17,20,23,40)(18,25)(19,31)(21,38,44,68)(22,39,42,65)(24,45,47,56)(26,49)(27,50)(28,36)(29,51)(30,48,54,73)(32,57)(33,37)(34,58)(35,59)(41,69)(43,67,70,80)(46,71,60,76)(52,64)(53,74)(55,72,75,77)(61,66)(62,78)(63,79);(2,3)(7,9)(8,10)(11,16)(17,21,42,70,23,44,22,43)(18,26,28,52,29,53,27,41)(19,32,34,62,35,63,33,61)(20,38,65,80,40,68,39,67)(24,46,54,75,47,60,30,55)(25,49,36,64,51,74,50,69)(31,57,58,78,59,79,37,66)(45,71,73,77,56,76,48,72)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efi,d^2=ehi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126806);