Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(9,10,11,12)(13,14)(15,19)(16,24)(17,22)(18,21)(20,25)(23,28)(26,27);(2,3)(5,6)(7,8)(9,10)(11,12)(13,15,17,18)(14,19,22,21)(16,23,25,26)(20,27,24,28);(6,8)(10,12)(13,16,17,25)(14,20,22,24)(15,23,18,26)(19,27,21,28);(1,3,4,2)(10,12)(14,21)(15,18)(19,22)(20,28)(23,26)(24,27)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335108);