Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120186 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,5,7,8)(4,6)(9,10,14,20,15,19,31,34,16,21,32,38,13,27,23,39)(11,22,26,30,25,35,40,28,24,17,33,29,18,36,37,12);(1,3)(2,5,6,8)(4,7)(9,11,23,37,13,18,32,33,16,24,31,40,15,25,14,26)(10,17,34,12,27,35,20,29,21,22,39,28,19,36,38,30);(9,12,13,29,16,28,15,30)(10,18,19,11,21,25,27,24)(14,22,23,36,32,17,31,35)(20,37,34,26,38,40,39,33)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ch,d^b=dgi,e^b=ei,f^b=fg,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49165);