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×#18347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3536 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);(3,4,7,12)(5,13,14,27)(6,17,20,11)(8,10,21,26)(9,18,24,15)(16,28,30,29)(19,32,34,25)(22,31,23,33);(3,5)(4,9)(6,18)(7,14)(8,16)(10,23)(11,27)(12,24)(13,17)(15,20)(19,33)(21,30)(22,26)(25,29)(28,32)(31,34);(6,19)(11,25)(13,28)(15,31)(17,32)(18,33)(20,34)(27,29);(4,10)(5,14)(9,22)(11,25)(12,26)(13,29)(15,18)(16,30)(17,32)(23,24)(27,28)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254103);