Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#23381 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 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)(6,7)(8,10)(11,12,14,16)(13,15,18,17)(19,20,23,26,22,24,29,31)(21,25,28,32,27,30,33,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,16)(15,18)(20,24)(21,27)(26,31)(28,33);(4,7)(8,9)(11,13,14,18)(12,15,16,17)(19,21,22,27)(20,25,24,30)(23,28,29,33)(26,32,31,34);(3,5,6,10)(4,8,7,9)(12,16)(15,17)(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, 6314932);