Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C23 | C2×#104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,21,16,20,13,33,17,23,32,19,14,22,12,30)(11,24,27,36,28,38,25,18,29,34,40,35,26,39,31,37);(2,5)(6,7)(9,11,16,28,17,29,14,26)(10,18)(12,31,15,27,13,25,32,40)(19,24)(20,35)(21,34)(22,36)(23,37)(30,38)(33,39);(1,3,4,8)(2,6,5,7)(9,12,14,32,17,13,16,15)(10,19,20,30,23,21,22,33)(11,25,26,27,29,31,28,40)(18,34,35,39,37,24,36,38)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49088);