Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11176 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2578 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);(3,4)(5,6)(7,8)(9,19,22,14)(10,23,26,34)(11,17)(12,18)(13,16)(15,25,32,28)(20,31)(21,33,27,30)(24,35)(29,36)(37,38);(7,9)(8,14)(10,24)(11,21)(12,27)(13,22)(15,31)(16,19)(17,30)(18,33)(20,25)(23,29)(26,36)(28,37)(32,38)(34,35);(4,6)(7,10)(8,15,16,32)(9,20)(11,25)(12,28)(13,26)(14,29,19,35)(17,23,18,34)(21,36)(22,37)(24,27)(30,38,33,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265123);