Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,5)(7,8,12,17)(9,13,21,30)(10,14,25,33)(11,26,24,16)(15,34,32,23)(18,28,20,29)(19,37,22,31)(27,38,35,36);(3,4)(5,6)(7,9)(8,13)(10,22)(11,18)(12,21)(14,31)(15,35)(16,29)(17,30)(19,25)(20,24)(23,38)(26,28)(27,32)(33,37)(34,36);(3,5,6,4)(8,14)(9,18)(10,23)(11,24)(13,27)(15,16)(17,33)(19,22)(20,21)(25,34)(26,32)(28,37)(29,31)(30,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406214);