Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#10866 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2162 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);(7,8)(9,17,20,13)(10,21)(11,24)(12,15)(14,22)(16,25)(18,31,33,35)(19,27,34,30)(23,36)(26,32,29,38)(28,37);(3,4,5,6)(7,9)(8,13)(10,18)(11,19)(12,20)(14,26)(15,17)(16,27)(21,35)(22,32)(23,33)(24,30)(25,34)(28,29)(31,36)(37,38);(7,10,11,22,12,23,25,37)(8,14,16,21,15,28,24,36)(9,18,19,32,20,33,34,38)(13,26,27,35,17,29,30,31)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260857);