Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#18370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3314 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)(5,7)(6,18)(11,27)(12,25)(13,22)(14,26)(16,31)(17,23)(19,34)(20,33)(21,30)(24,29)(28,32);(3,4)(5,10)(6,19)(7,22)(8,13)(9,14)(11,21)(12,15)(16,32)(17,26)(18,28)(20,24)(23,25)(27,31)(29,30)(33,34);(3,5,15,23)(4,10,12,25)(6,16,30,31)(7,9,17,8)(11,24,28,34)(13,22,14,26)(18,33,21,20)(19,32,29,27);(3,6,15,30)(4,11,12,28)(5,16,23,31)(7,20,17,33)(8,21,9,18)(10,24,25,34)(13,19,14,29)(22,32,26,27)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254129);