Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,7,8)(5,6,9,10)(12,16)(14,17)(20,23)(21,25);(4,8)(5,9)(11,12)(13,17)(14,18)(15,16)(20,23)(24,26);(12,16)(13,18)(19,20,22,23)(21,24,25,26);(3,5)(4,6)(7,9)(8,10)(11,13)(12,14)(15,18)(16,17)(19,21)(20,24)(22,25)(23,26);(3,6)(4,5)(7,10)(8,9)(11,14)(12,13)(15,17)(16,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008590);