Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(8,10)(12,15)(13,17)(19,20,24,28)(21,22,26,25)(23,31,32,27)(29,33,34,30);(11,12,14,15)(13,16,17,18)(21,26)(22,25)(29,34)(30,33);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,26)(20,25,28,22)(23,29,32,34)(27,33,31,30);(3,5,6,9)(4,8,7,10)(11,13,14,17)(12,16,15,18)(19,22,24,25)(20,26,28,21)(23,30,32,33)(27,34,31,29);(19,23,24,32)(20,27,28,31)(21,29,26,34)(22,30,25,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541994);