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×#36295 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(20,24)(21,26);(3,4)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,18,17)(19,20,23,24)(21,25,26,22);(3,4,6,8)(5,7,10,9)(11,12)(13,17)(14,16)(15,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,9)(6,8)(7,10)(11,13)(12,15)(14,18)(16,17)(19,22,23,25)(20,21,24,26);(3,5)(4,7)(6,10)(8,9)(11,12)(13,17)(14,16)(15,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537142);