Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,13,11,12,16,21,19,14,17,23,20,22,15,24,18);(1,2)(3,5)(4,6)(7,8)(9,11,13,16,12,19,21,17,14,20,23,15,22,18,24,10);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=eg,d^c=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43132);