Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#39363 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#550 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11250 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);(8,10)(11,12,16,15)(13,19,21,17)(14,22,23,18)(20,24,26,25);(4,5)(12,15)(17,19)(18,22)(24,25);(3,4,6,5)(7,8,9,10)(11,13)(12,17)(14,20)(15,19)(16,21)(18,24)(22,25)(23,26);(11,14,16,23)(12,18,15,22)(13,20,21,26)(17,24,19,25);(3,5,6,4)(11,15,16,12)(13,19,21,17)(14,22,23,18)(20,25,26,24)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543930);