Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,11,14)(9,15,17,12)(10,13,18,20)(16,21,22,19)(23,24,28,36,27,32,26,31)(25,29,33,30,35,38,34,37);(3,4,5,6)(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21)(23,25)(24,29)(26,34)(27,35)(28,33)(30,36)(31,37)(32,38);(4,6)(10,18)(13,20)(16,22)(19,21)(23,25)(24,30)(26,33)(27,35)(28,34)(29,36)(31,38)(32,37);(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)(23,26,27,28)(24,31,32,36)(25,33,35,34)(29,30,38,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343565);