Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1745 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,13,18,12,15,21,24,14,19,23,16,22,17,11,20);(1,2,3,5,4,6,7,8)(10,15,19,17)(16,20,18,24);(9,11,12,13,14,21,22,23)(10,16,17,24,19,18,15,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35408);