Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,13,19)(9,20,24,31)(10,14,11,27)(12,29,26,18)(15,21,33,22)(16,28,17,25)(23,30,36,34)(32,38,35,37);(8,14)(9,21)(10,25)(11,28)(12,26)(16,18)(17,29)(19,27)(20,34)(22,24)(23,38)(30,31)(32,35)(36,37);(4,6)(7,9,10,21,26,36,28,38,13,24,11,22,12,23,25,37)(8,15,16,31,29,35,27,34,19,33,17,20,18,32,14,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406352);