Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142216 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
971 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,15)(14,16)(18,21)(22,24);(1,3,4,8)(2,5,6,7)(19,23);(10,13)(11,16)(14,15)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,e^c=eg,h^c=hi,f^e=fi50 conjugacy classes — view the full character table.
| maximal quotient of | #72267773, #72267776, #72267777 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51628);