Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23601 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3158 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);(7,8,11,16,13,17,12,26)(9,18,20,29,22,35,21,14)(10,15,27,32,25,33,24,31)(19,34,37,28,36,30,23,38);(3,4)(5,6)(7,9,12,21,13,22,11,20)(8,14,16,18,17,29,26,35)(10,23,24,36,25,37,27,19)(15,30,32,38,33,34,31,28);(4,6)(7,10)(8,15)(9,19)(11,24)(12,27)(13,25)(14,28)(16,31)(17,33)(18,34)(20,23)(21,37)(22,36)(26,32)(29,38)(30,35);(7,11,13,12)(8,16,17,26)(9,20,22,21)(10,24,25,27)(14,18,29,35)(15,31,33,32)(19,23,36,37)(28,34,38,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339311);