Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 182 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)(3,7)(4,6)(5,8)(9,10,12,18,13,15,21,22,14,16,20,23,11,17,19,24);(1,3)(2,5)(4,8)(6,7)(10,15,16,17)(12,20)(18,24,23,22)(19,21);(3,8)(5,7)(10,16)(12,20)(15,17)(19,21);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=ceh,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403822);