Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,13)(10,16)(12,18)(15,17)(20,25)(21,29)(22,23)(27,28)(30,31)(32,33);(3,5,7,12)(4,8,9,17)(6,15,16,13)(10,18,14,11)(19,20,24,28)(21,26,30,34)(22,25,23,27)(29,32,31,33);(19,21,24,30)(20,26,28,34)(22,29,23,31)(25,32,27,33);(4,9)(5,12)(10,14)(13,15)(19,22,24,23)(20,27,28,25)(21,29,30,31)(26,33,34,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255448);