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×#39360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11034 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)(12,16)(13,19)(18,21)(20,25);(4,5)(11,12)(13,17)(14,21)(15,16)(18,22)(19,23)(20,26)(24,25);(3,4,6,5)(7,8,9,10);(11,13,15,19)(12,17,16,23)(14,20,22,25)(18,24,21,26);(3,5,6,4)(11,14,15,22)(12,18,16,21)(13,20,19,25)(17,24,23,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543911);