Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1556 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 4 generators on 24 points:
(1,2)(3,4)(6,7)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(5,6)(7,8)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(2,4)(5,7,8,6)(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(10,15)(14,22)(16,21)(18,23)Polycyclic presentation:
2^9|c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309669, #72309679, #72309704, … |
| maximal normal subgroup of | #72309669, #72309679, #72309704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406259);