Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3160 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(5,7,8,12)(6,9,10,11);(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482461);