Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27359 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12,15,16)(13,18,14,17)(19,20)(21,24)(22,25)(23,26)(27,34)(28,32)(29,31)(30,33);(3,5,7,6)(4,8,9,10)(12,16)(17,18)(19,21,22,28)(20,24,25,32)(23,27,30,29)(26,34,33,31);(3,6,7,5)(4,8,9,10)(11,13,15,14)(12,17,16,18)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34);(4,9)(8,10)(11,14,15,13)(12,18,16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357475);