Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#23312 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C2×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,26,22,24,29,31)(21,25,28,32,27,30,33,34);(3,5)(4,7)(6,10)(8,9)(11,13)(12,15)(14,17)(16,18)(20,24)(21,27)(26,31)(28,33);(5,10)(7,9)(12,16)(13,17)(19,21)(20,25)(22,27)(23,28)(24,30)(26,32)(29,33)(31,34);(4,8)(5,10)(12,16)(15,18)(20,24)(21,27)(26,31)(28,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314650);