Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
2130 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2130 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,23,21,24,19,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13,14,15)(12,16);(2,6)(5,7)(17,19,21,20)(18,22,24,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,e^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2311);