Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C26×C43 |
| direct product | C4×#2275 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C2 | C22×C43 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4);(5,6,8,7)(13,14)(15,19)(16,18)(17,20);(9,10,11,12)(13,15)(14,17)(16,20)(18,19);(5,7,8,6)(15,20)(17,19);(14,18)(17,19)272 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420526);