Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8738 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1817 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
808 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 808 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,14,17,21,24,13,15,20,22,11,16,19,23);(10,15)(12,20)(16,17)(19,21);(1,3)(2,5)(4,8)(6,7)(10,16,15,17)(18,23,22,24)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^b=fi,f^c=fgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219601, #72224579, #72239364, … |
| maximal normal subgroup of | #72219601, #72224579, #72239364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42353);