Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2510 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);(3,4,6,5)(7,8,12,17,13,18,11,16)(9,19,22,14,23,30,21,31)(10,24,27,33,28,15,26,34)(20,35,36,38,37,32,25,29);(3,5,6,4)(7,9)(8,14)(10,25)(11,22)(12,21)(13,23)(15,32)(16,30)(17,19)(18,31)(20,26)(24,35)(27,37)(28,36)(29,33)(34,38);(7,9)(8,14)(10,20)(11,21)(12,22)(13,23)(15,29)(16,19)(17,30)(18,31)(24,38)(25,26)(27,36)(28,37)(32,33)(34,35);(7,10,11,26,13,28,12,27)(8,15,16,33,18,24,17,34)(9,20,21,25,23,37,22,36)(14,29,19,32,31,38,30,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289198);