Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476654 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2,4,6)(3,5,7,8)(9,10,12,15,14,18,13,17)(11,16,19,23,21,24,20,22);(1,3)(2,5)(4,7)(6,8)(10,15,18,17)(16,23,24,22);(3,7)(5,8)(10,15)(11,19)(12,13)(16,24)(17,18)(20,21);(9,11)(10,16,18,24)(12,20)(13,19)(14,21)(15,22,17,23)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386501);