Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#8932 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,6,9)(5,7,10,8)(11,12,16,19,17,14,15,20)(13,21,23,18,25,26,24,22);(11,13,15,24,17,25,16,23)(12,18,20,21,14,22,19,26);(4,7)(5,10)(8,9)(11,14)(12,17)(13,22)(15,19)(16,20)(18,25)(21,23)(24,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100521);