Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112870 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 413 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5,7,8)(4,6)(9,10,14,20,16,19,34,36,15,21,33,39,13,32,23,24)(11,22,27,40,26,30,18,38,28,12,29,31,25,17,35,37);(1,3)(2,5,6,8)(4,7)(9,11,23,35,13,25,33,29,15,28,34,18,16,26,14,27)(10,17,36,40,32,12,20,37,21,30,24,31,19,22,39,38);(9,12,13,30,15,22,16,17)(10,18,19,29,21,35,32,27)(11,24,26,20,28,36,25,39)(14,31,23,38,33,40,34,37)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=i,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=fgi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47858);