Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30284 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
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)(23,24,26,30,27,29,35,37)(25,31,32,36,33,38,34,28);(3,4,5,6)(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21);(4,6)(10,18)(13,20)(16,22)(19,21)(23,25,27,33)(24,28,29,36)(26,34,35,32)(30,38,37,31);(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)(24,29)(28,36)(30,37)(31,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343562);