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×#17621 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9,11,24)(5,14)(6,18)(10,26,27,32)(12,25,13,19)(15,17)(20,22)(23,29,34,28);(3,4,7,12)(5,10,14,28)(6,19,18,24)(8,13,21,11)(9,22,25,20)(15,27,17,29)(16,32,30,34)(23,31,26,33);(3,5)(4,10)(6,16)(7,14)(8,17)(9,23)(11,27)(12,28)(13,29)(15,21)(18,30)(19,32)(20,33)(22,31)(24,34)(25,26);(5,15)(10,27)(14,17)(16,31)(23,34)(26,32)(28,29)(30,33)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250659);