Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60308 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3483 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7118 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)(5,9)(6,8)(7,10)(11,12,16,19,15,14,18,23)(13,20,22,26,21,17,24,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,20)(15,21)(16,22)(18,24)(19,25)(23,26);(5,10)(7,9)(11,13)(12,17)(14,20)(15,21)(16,22)(18,24)(19,25)(23,26);(4,8)(7,9)(13,21)(17,20)(22,24)(25,26);(11,14,16,23,15,12,18,19)(13,20,22,26,21,17,24,25)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481664);