Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11241 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 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);(7,8)(9,17)(10,14,22,28)(11,23)(12,16)(13,19)(15,24)(18,29,32,38)(20,30)(21,34,35,27)(25,36,37,31)(26,33);(3,4)(5,6)(7,9,11,19,12,20,24,33)(8,13,15,17,16,26,23,30)(10,18,21,31,22,32,35,36)(14,25,27,29,28,37,34,38);(4,6)(7,10,11,21,12,22,24,35)(8,14,15,27,16,28,23,34)(9,18,19,31,20,32,33,36)(13,25,17,29,26,37,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265254);