Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#16971 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2726 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);(4,7)(8,10)(11,12,14,16)(13,15,17,18)(19,20,23,24)(21,22,26,25)(27,28)(29,33)(30,32)(31,34);(4,7)(8,10)(20,24)(21,26)(27,29,32,31)(28,33,30,34);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,26)(20,25,24,22)(27,30,32,28)(29,33,31,34);(3,5,6,9)(4,8,7,10)(11,13,14,17)(12,15,16,18)(19,22,23,25)(20,21,24,26)(27,31,32,29)(28,33,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519015);