Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#33459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1986 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);(3,4,6,5)(7,8)(9,15)(10,18)(11,25)(12,16)(13,19)(14,21)(17,27)(20,34)(22,33)(23,31)(24,36)(26,32)(28,35)(29,38)(30,37);(7,9)(8,14)(10,20)(11,26)(12,23)(13,22)(15,25)(16,29)(17,31)(18,32)(19,30)(21,27)(24,37)(28,38)(33,35)(34,36);(3,5,6,4)(7,9)(8,15)(10,23)(11,21)(12,20)(13,22)(14,25)(16,34)(17,32)(18,31)(19,33)(24,38)(26,27)(28,37)(29,36)(30,35);(7,10,13,12)(8,16,19,18)(9,20,22,23)(11,24,28,27)(14,29,30,32)(15,31,33,34)(17,35,36,25)(21,26,37,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529805);