Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#18259 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(7,10)(11,12,17,14)(13,21,24,26)(15,20,16,19)(18,25,22,23);(3,4)(5,7)(6,8)(9,10)(11,13)(12,18)(14,22)(15,23)(16,25)(17,24)(19,21)(20,26);(3,5)(4,7)(6,9)(8,10)(11,14,17,12)(13,22,24,18)(15,19,16,20)(21,25,26,23);(11,15,17,16)(12,19,14,20)(13,23,24,25)(18,21,22,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253751);