Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#33433 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,10,15,11,16,20,28,12,14,21,26,22,27,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(4,6)(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38);(7,9,10,18,11,19,20,31,12,17,21,29,22,30,32,37)(8,13,15,24,16,25,28,35,14,23,26,33,27,34,36,38);(8,14)(13,23)(15,26)(16,27)(24,33)(25,34)(28,36)(35,38)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529766);