Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,14)(12,13)(15,16)(17,18,19,22,20,23,24,21);(5,6,7,8)(10,13)(14,16);(1,2,3,4)(18,21)(19,24)(22,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^c=fi,g^c=gi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261345);