Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13304 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(9,10,15,21)(11,23,26,19)(12,22,16,18)(13,28,31,35)(14,32,29,38)(17,30,36,25)(20,27,40,33)(24,39,34,37);(5,6)(7,8)(9,11,15,26)(10,17,21,36)(12,24,16,34)(13,29,31,14)(18,35,22,28)(19,32,23,38)(20,37,40,39)(25,27,30,33);(2,4)(6,8)(9,12,15,16)(10,18,21,22)(11,24,26,34)(13,30,31,25)(14,27,29,33)(17,35,36,28)(19,39,23,37)(20,38,40,32);(2,4)(9,13,27,34,16,30,14,11,15,31,33,24,12,25,29,26)(10,19,38,17,22,39,20,35,21,23,32,36,18,37,40,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416803);