Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144627 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
705 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 705 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,5)(6,8)(9,10,13,16)(11,14,18,23)(12,19)(15,21)(17,24)(20,22);(3,7)(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=dg,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,c^b=cf,e^b=eh,f^b=fg,d^c=dg,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72232798, #72233000, #72233146, … |
| maximal normal subgroup of | #72232798, #72233000, #72233146, … |
| maximal quotient of | #72251756, #72251757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51850);