Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8,10,12)(9,11,13,14)(16,21)(19,27)(22,23)(25,26);(7,8)(9,11)(10,12)(13,14)(15,16)(17,23)(18,22)(19,24)(20,21)(25,30)(26,29)(27,28);(3,4,5,6)(7,8)(9,11)(10,12)(13,14)(15,17,20,18)(16,22,21,23)(19,25,27,26)(24,29,28,30);(7,9,10,13)(8,11,12,14)(15,18,20,17)(16,23,21,22)(19,26,27,25)(24,30,28,29);(15,19,20,27)(16,24,21,28)(17,25,18,26)(22,29,23,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535798);