Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,6,8,7,10,5,9)(11,12,16,20,17,19,15,13)(14,21,22,18,23,26,24,25);(4,8)(5,6)(9,10)(11,13,16,19,17,20,15,12)(14,21,22,25,23,26,24,18);(4,8)(5,6)(9,10)(11,14)(12,18)(13,21)(15,24)(16,22)(17,23)(19,25)(20,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106120);