Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3310 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)(5,14)(6,16)(7,19)(11,24)(12,20)(13,26)(15,29)(17,33)(18,21)(22,31)(25,27)(28,34)(30,32);(3,4)(5,11)(6,17)(7,20)(8,13)(9,12)(10,19)(14,27)(15,30)(16,32)(18,25)(21,33)(22,34)(23,28)(24,29)(26,31);(3,5,9,6)(4,11,12,17)(7,21,19,29)(8,14,23,16)(10,24,20,33)(13,27,28,32)(15,31,18,22)(25,34,30,26);(3,6,9,5)(4,11,12,17)(7,18,19,15)(8,16,23,14)(10,25,20,30)(13,27,28,32)(21,22,29,31)(24,34,33,26)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254110);