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×#11150 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2538 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)(9,18)(10,22,26,36)(11,27)(12,16)(13,17)(14,20)(15,25,33,24)(19,29,23,32)(21,30)(28,31)(34,35,38,37);(3,4)(5,6)(7,9)(8,14)(10,23)(11,28)(12,20)(13,21)(15,32)(16,30)(17,31)(18,27)(19,26)(22,34)(24,35)(25,37)(29,33)(36,38);(4,6)(7,10,11,24,13,26,12,25)(8,15,16,22,17,33,27,36)(9,19,20,35,21,23,28,37)(14,29,18,34,31,32,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265104);