Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25751 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,5)(7,8)(9,18)(10,22)(11,16)(12,26)(13,17)(14,20)(15,25)(19,36)(21,30)(23,29)(24,34)(27,31)(28,33)(32,37)(35,38);(3,4,6,5)(7,9,11,20)(8,14,16,18)(10,23,24,38)(12,21,13,27)(15,32,33,36)(17,30,26,31)(19,28,37,25)(22,35,34,29);(7,10,12,25,11,24,13,28)(8,15,17,22,16,33,26,34)(9,19,21,38,20,37,27,23)(14,29,30,32,18,35,31,36);(3,5,6,4)(8,16)(10,24)(14,18)(17,26)(19,37)(23,38)(25,28)(30,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350504);