Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C2×#34268 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,6)(7,8,10,9)(11,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,25)(22,26);(7,8,10,9)(12,17)(14,18)(19,23)(22,25);(3,4)(5,6)(7,9,10,8)(11,13)(12,18)(14,17)(15,20)(16,21)(19,25)(22,23)(24,26);(11,14,16,18)(12,13,17,21)(15,22,24,25)(19,20,23,26);(11,15,16,24)(12,19,17,23)(13,20,21,26)(14,22,18,25)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532759);