Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1344 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)(7,8,12,17,14,10,13,25)(9,18,22,35,23,34,30,36)(11,26,28,24,19,37,29,16)(15,31,32,21,20,38,33,27);(7,9,19,21,13,22,28,38,14,23,11,27,12,30,29,31)(8,15,16,34,17,33,26,35,10,20,24,18,25,32,37,36);(4,6)(7,10)(8,14)(9,20)(11,16)(12,17)(13,25)(15,23)(18,21)(19,24)(22,32)(26,29)(27,34)(28,37)(30,33)(31,35)(36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401853);