Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,6)(4,5)(7,8)(9,10,12,17,14,18,13,22)(11,15,19,16,21,24,20,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(11,19)(12,13)(16,17)(18,24)(20,21)(22,23)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72264968, #72265003, #72265006, … |
| maximal normal subgroup of | #72264968, #72265003, #72265006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42490);