Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1600 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 190 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:
(1,2)(3,7,8,5)(4,6)(9,10)(11,15,16,13)(12,14)(17,18,19,21,20,22,23,24);(1,3,4,8)(2,5,6,7);(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266345, #72266347 |
| maximal normal subgroup of | #72266345, #72266347 |
| maximal quotient of | #72266347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4265);