Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102164 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| prime radical | #33301 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
724 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,11,14,12,15,16,13)(18,21)(19,23)(22,24);(1,2,6,8,4,7,3,5)(10,13)(11,16)(14,15)(18,22)(21,24);(17,18,19,21,20,22,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=e,d^2=gh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9471);