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×#17629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3506 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3272 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);(4,10,12,25)(5,15)(7,21)(11,24,28,27)(13,26,14,22)(16,31)(17,18)(29,34,30,33);(3,4)(5,10)(6,11)(7,22)(8,13)(9,14)(12,23)(15,26)(16,24)(17,27)(18,33)(19,29)(20,30)(21,25)(28,32)(31,34);(3,5,9,7)(4,10,14,22)(6,17,20,31)(8,15,23,21)(11,27,30,34)(12,25,13,26)(16,19,18,32)(24,29,33,28);(3,6,9,20)(4,11,14,30)(5,16,7,18)(8,19,23,32)(10,24,22,33)(12,28,13,29)(15,31,21,17)(25,27,26,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250669);