Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1974 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)(7,8,13,10)(9,17,21,36)(11,24,28,35)(12,23,26,16)(14,31,18,20)(15,30,22,27)(19,37,29,33)(25,34,38,32);(7,9,13,21)(8,14,10,18)(11,25,28,38)(12,29,26,19)(15,33,22,37)(16,34,23,32)(17,35,36,24)(20,30,31,27);(4,6)(7,10,13,8)(9,18,21,14)(11,22,28,15)(12,23,26,16)(17,20,36,31)(19,34,29,32)(24,27,35,30)(25,37,38,33);(7,11)(8,15)(9,19)(10,22)(12,30)(13,28)(14,32)(16,35)(17,36)(18,34)(20,31)(21,29)(23,24)(26,27)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530778);