Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C8×#400 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(21,22,23,25,24,26,27,28);(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(17,20)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266554);