Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C24×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C2×C43 |
| Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(17,18)(19,22)(20,21)(23,24);(9,10,11,12)(18,21)(22,24);(2,4)(13,14,15,16);(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=e,b^a=be,c^a=cf224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60907);