Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8119 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,5,6,9,8,10)(11,12)(13,21,23,17)(14,18)(15,20)(16,19)(22,26,24,25);(3,5,8,9,6,10,7,4)(11,13,15,24,16,23,14,22)(12,17,20,26,19,21,18,25);(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88587);