Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#20514 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,12,17,13,15,11,26)(9,18,20,14,21,30,27,28)(10,22,25,37,23,16,24,31)(19,33,35,36,34,32,38,29);(3,4,6,5)(7,9)(8,14)(10,19)(11,27)(12,20)(13,21)(15,28)(16,29)(17,30)(18,26)(22,36)(23,34)(24,38)(25,35)(31,33)(32,37);(3,5,6,4)(8,15)(10,23)(14,28)(17,26)(18,30)(19,34)(24,25)(35,38);(7,10,12,25,13,23,11,24)(8,16,17,31,15,22,26,37)(9,19,20,35,21,34,27,38)(14,29,30,33,28,36,18,32)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277285);