Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C8 · Schur multiplier (H2) = C214 |
| direct product | C2×C8×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C2 | C25×C8 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C22×2−1+4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8274 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);(3,4,5,6,7,8,9,10);(12,16)(14,22)(17,19)(20,26);(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24);(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)272 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481734);