Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#30233 |
| 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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 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)(5,7)(6,8)(9,10)(11,12,15,13,16,17,14,18)(19,20,23,24)(21,25,28,31)(22,26,30,32)(27,33,29,34);(3,5,6,9)(4,7,8,10)(11,13,14,12,16,18,15,17)(20,24)(25,31)(26,32)(33,34);(4,8)(7,10)(12,13)(14,15)(17,18)(19,21,23,28)(20,25,24,31)(22,29,30,27)(26,34,32,33);(12,17)(13,18)(19,22,23,30)(20,26,24,32)(21,27,28,29)(25,33,31,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343535);