Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6294 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);(3,4,7,5)(6,8,10,9)(11,12)(13,14)(15,18)(16,17);(3,4,7,5)(6,9,10,8)(12,17)(13,15)(19,20,24,22)(21,26,23,25);(4,5)(8,9)(11,13,16,15)(12,14,17,18)(19,21,24,23)(20,25,22,26);(3,5,7,4)(6,8,10,9)(11,14)(12,13)(15,17)(16,18)(19,22,24,20)(21,25,23,26);(3,6,7,10)(4,8,5,9)(11,15,16,13)(12,18,17,14)(19,23,24,21)(20,26,22,25)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537465);