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×#13394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,14,20)(9,21,24,33)(10,25,22,15)(11,19,13,17)(12,28,31,37)(16,34,36,26)(18,27,38,30)(23,35,29,32);(3,4)(5,6)(7,9)(8,15)(10,11)(12,29)(13,22)(14,24)(16,28)(17,33)(18,35)(19,21)(20,25)(23,31)(26,30)(27,34)(32,38)(36,37);(4,6)(7,10,11,9,14,22,13,24)(8,16,17,32,20,36,19,35)(12,29,27,26,31,23,30,34)(15,18,21,37,25,38,33,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406227);