Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,7)(9,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,28)(22,26)(23,25)(24,27);(1,2,4,3)(5,7,8,6)(13,15,16,22)(14,18,19,26)(17,21,24,23)(20,28,27,25);(1,2,4,3)(6,7)(14,19)(15,21)(17,24)(18,25)(22,23)(26,28);(1,3,4,2)(9,11,12,10)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285165);