Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3365 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 3365 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(2,4)(5,6)(7,8);(6,8)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(2,4)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313958, #72314006, #72314039 |
| maximal normal subgroup of | #72313958, #72314006, #72314039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281219);