Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1556 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);(3,4)(5,6)(7,8)(9,20)(10,26)(11,29)(12,18)(13,32)(14,19)(15,22)(16,25)(17,21)(23,35)(24,34)(27,33)(28,37)(30,36)(31,38);(4,6)(7,9,14,25)(8,15,19,33)(10,11,28,31)(12,24,13,23)(16,29,20,38)(17,35,36,34)(18,37,32,26)(21,22,30,27);(7,10,14,28)(8,16,19,20)(9,21,25,30)(11,23,31,24)(12,22,13,27)(15,17,33,36)(18,35,32,34)(26,38,37,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406054);