Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46346 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(12,16)(17,18)(19,20,24,26)(21,27,29,22)(23,31,32,25)(28,30,34,33);(4,6)(5,9)(19,21,24,29)(20,22,26,27)(23,28,32,34)(25,30,31,33);(3,4,7,6)(5,8,9,10)(11,12)(13,17)(14,18)(15,16);(3,5,7,9)(4,8,6,10)(11,13,15,14)(12,17,16,18)(19,22)(20,21)(23,30)(24,27)(25,28)(26,29)(31,34)(32,33);(3,6,7,4)(5,10,9,8)(11,14,15,13)(12,18,16,17)(19,23,24,32)(20,25,26,31)(21,28,29,34)(22,30,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610249);