Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6046 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);(4,6)(5,9)(12,17)(13,15)(19,20)(21,25)(22,23)(24,26);(3,4,7,6)(5,10,9,8)(11,12,16,17)(13,14,15,18)(20,23)(21,26);(5,9)(8,10)(11,13,16,15)(12,18,17,14)(19,20)(21,24)(22,23)(25,26);(3,5,7,9)(4,8,6,10)(11,14,16,18)(12,13,17,15)(19,21,22,26)(20,24,23,25);(3,6,7,4)(5,10,9,8)(11,15,16,13)(12,18,17,14)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000080);