Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
918 conjugacy classes of subgroups; only the 394 normal subgroups are shown.
This group has 918 conjugacy classes of subgroups — too many to draw. Only its 394 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) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(10,14)(11,12)(15,16)(17,18,19,22,20,21,23,24);(2,4)(9,10,12,16,13,15,11,14);(5,6,7,8)(10,15)(14,16)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=dh,f^a=fh,c^b=ch128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12470);