Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3251 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,8)(5,6)(9,10)(11,12)(13,21)(14,23)(15,25)(16,19)(17,22)(18,24)(20,26);(3,4)(5,8)(6,10)(7,9)(11,13,16,18)(12,17,19,14)(15,23,26,22)(20,24,25,21);(3,5,7,6)(4,8,9,10)(11,14,16,17)(12,18,19,13)(15,21,26,24)(20,22,25,23);(4,9)(8,10)(12,19)(13,18)(20,25)(22,23)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255955);