Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34530 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,13)(10,17)(11,12)(14,18)(15,21)(16,19)(20,22)(23,24,28,31)(25,26,33,29)(27,35,36,30)(32,37,38,34);(8,12)(13,19)(14,17)(20,21)(23,25,28,33)(24,29,31,26)(27,32,36,38)(30,37,35,34);(3,4,6,5)(7,9,11,16)(8,13,12,19)(10,15,18,22)(14,20,17,21)(23,25)(24,29)(26,31)(27,32)(28,33)(30,37)(34,35)(36,38);(7,8)(9,13)(10,14)(11,12)(15,20)(16,19)(17,18)(21,22)(23,26,28,29)(24,25,31,33)(27,34,36,37)(30,32,35,38);(3,5,6,4)(7,10)(8,14)(9,15)(11,18)(12,17)(13,20)(16,22)(19,21)(23,27,28,36)(24,30,31,35)(25,32,33,38)(26,34,29,37)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533350);