Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
| direct product | C2×#52307 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7442 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);(3,4,7,10)(5,8,6,9)(12,15)(13,17)(19,20,23,24)(21,25,22,26);(3,4)(5,8)(6,9)(7,10)(11,12,14,15)(13,16,17,18)(20,24)(21,22);(3,5,7,6)(4,8,10,9)(11,13,14,17)(12,16,15,18)(19,21,23,22)(20,25,24,26);(3,5,7,6)(4,9,10,8)(12,15)(16,18)(19,22,23,21)(20,25,24,26);(3,6,7,5)(4,9,10,8)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540832);