Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11236 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #394 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1858 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)(10,23)(11,17)(12,16)(13,18)(14,21)(15,25)(20,30)(22,33)(24,32)(26,31)(27,29)(28,36)(34,35)(37,38);(7,9,11,21,13,22,12,27)(8,14,16,19,18,29,17,33)(10,20,24,35,26,36,25,37)(15,30,31,38,32,28,23,34);(7,10,11,24,13,26,12,25)(8,15,16,31,18,32,17,23)(9,20,21,35,22,36,27,37)(14,28,19,34,29,30,33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265244);