Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60981 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3188 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,30,22,26)(23,29,25,33)(27,31,34,32);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21)(20,26)(22,24)(23,32)(25,31)(27,33)(28,30)(29,34);(5,12)(8,11)(13,18)(15,17)(19,22,24,21)(20,27,28,34)(23,32,25,31)(26,29,30,33);(4,9)(8,11)(10,16)(15,17)(20,28)(26,30)(27,34)(29,33);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(19,23,24,25)(20,29,28,33)(21,31,22,32)(26,27,30,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483019);