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×#36165 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6266 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)(5,8)(6,9)(7,10)(11,12,15,17)(13,14,16,18)(21,25)(23,26);(3,5,6,10)(4,7,9,8)(11,13)(12,14)(15,16)(17,18)(19,20,22,24)(21,23,25,26);(3,4,6,9)(5,8,10,7)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,25)(24,26);(3,4)(5,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,18);(3,5)(4,8)(6,10)(7,9)(11,14)(12,16)(13,17)(15,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536997);