Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#33527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,18)(10,15)(11,24)(12,22)(13,17)(14,20)(16,26)(19,35)(21,36)(23,33)(25,29)(27,34)(28,32)(30,38)(31,37);(7,9,13,21)(8,14,17,31)(10,19,12,28)(11,25,27,38)(15,29,22,30)(16,32,33,35)(18,34,36,24)(20,23,37,26);(4,6)(7,8)(9,14)(10,22)(11,16)(12,15)(13,17)(18,20)(19,30)(21,31)(23,34)(24,26)(25,32)(27,33)(28,29)(35,38)(36,37);(7,10,13,12)(8,15,17,22)(9,19,21,28)(11,23,27,26)(14,29,31,30)(16,24,33,34)(18,35,36,32)(20,25,37,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530014);