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×#11177 |
| 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);(7,8,12,18,13,16,11,17)(9,19,21,31,23,14,22,30)(10,24,26,33,28,34,27,15)(20,36,37,35,38,32,25,29);(3,4)(5,6)(7,9)(8,14)(10,25)(11,21)(12,22)(13,23)(15,32)(16,19)(17,30)(18,31)(20,27)(24,35)(26,38)(28,37)(29,34)(33,36);(4,6)(7,10)(8,15,16,33)(9,20)(11,26)(12,27)(13,28)(14,29,19,35)(17,24,18,34)(21,25)(22,37)(23,38)(30,32,31,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265125);