Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#33432 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| 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×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2106 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)(9,13,17,24,18,25,29,35,19,23,30,33,31,34,37,38);(4,6);(7,9,10,17,11,18,20,29,12,19,21,30,22,31,32,37)(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)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529765);