Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17760 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,10,7,8,13,15)(5,9,12,16,11,14,17,18)(20,24)(21,27)(22,29)(25,32)(26,30)(31,33);(4,8)(6,13)(9,14)(12,17)(19,20,23,26)(21,25,27,33)(22,30,29,24)(28,32,34,31);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21)(20,25)(22,28)(23,27)(24,31)(26,33)(29,34)(30,32);(5,11)(9,14)(12,17)(16,18)(21,27)(25,33)(28,34)(31,32)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251256);