Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11264 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1626 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);(4,5)(7,8,12,16,13,18,11,17)(9,19,21,14,23,30,22,31)(10,24,26,34,27,33,28,15)(20,32,37,38,25,29,36,35);(3,4,6,5)(7,9,11,22,13,23,12,21)(8,14,17,19,18,31,16,30)(10,25,26,36,27,20,28,37)(15,32,24,38,34,29,33,35);(3,5,6,4)(7,10,11,26,13,27,12,28)(8,15,16,33,18,34,17,24)(9,20,21,36,23,25,22,37)(14,29,19,35,31,32,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265473);