Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#22308 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,8)(6,10)(7,9)(11,12)(13,15)(14,17)(16,18)(19,20,24,27,23,26,28,21)(22,30,32,29,31,25,33,34);(3,5,7,6)(4,8,9,10)(11,13,15,12)(14,17,18,16)(19,21,28,26,23,27,24,20)(22,29,33,30,31,34,32,25);(3,6,7,5)(4,8,9,10)(12,13)(16,17)(19,22,23,31)(20,25,26,30)(21,29,27,34)(24,32,28,33);(11,14,15,18)(12,16,13,17)(22,31)(25,30)(29,34)(32,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283879);