Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#21545 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2722 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)(5,6)(7,8,12,16,13,17,11,26)(9,18,21,28,22,32,20,14)(10,15,25,30,23,29,24,36)(19,31,35,37,33,38,34,27);(7,9,11,20,13,22,12,21)(8,14,16,18,17,28,26,32)(10,19,24,34,23,33,25,35)(15,27,30,31,29,37,36,38);(4,6)(10,23)(15,29)(19,33)(24,25)(27,37)(30,36)(31,38)(34,35);(7,10,12,25,13,23,11,24)(8,15,16,30,17,29,26,36)(9,19,21,35,22,33,20,34)(14,27,18,31,28,37,32,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280090);