Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19618 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#127 |
| prime radical | C24 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,15)(13,18)(14,17)(19,20,23,26)(21,27,30,34)(22,31,28,25)(24,32,33,29);(3,5,7,15,8,13,6,14)(4,9,10,12,11,17,16,18);(5,13)(9,17)(12,18)(14,15)(19,21,28,32,23,30,22,29)(20,24,31,34,26,33,25,27);(3,6,8,7)(4,10,11,16)(5,14,13,15)(9,12,17,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264292);