Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 733 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,22,13,16,23,17)(11,18,21,15,19,14,20,24);(1,3,4,8)(2,5,6,7)(11,19)(14,18)(15,24)(20,21);(10,14)(11,20)(12,23)(15,22)(16,18)(17,24)(19,21);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=gh,b^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237246, #72239808, #72239816, … |
| maximal normal subgroup of | #72237246, #72239808, #72239816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412776);