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×#36605 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6282 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);(3,4,7,8)(5,6,9,10)(11,12,16,14)(13,17,15,18)(20,24)(25,26);(3,4)(5,6)(7,8)(9,10)(11,12,16,14)(13,18,15,17)(19,20,23,24)(21,25,22,26);(4,8)(5,9)(11,13,16,15)(12,17,14,18)(19,21,23,22)(20,25,24,26);(3,5)(4,6)(7,9)(8,10)(11,14,16,12)(13,17,15,18);(3,6)(4,9)(5,8)(7,10)(11,15,16,13)(12,18,14,17)(19,22,23,21)(20,26,24,25)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537515);