Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45998 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7446 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)(8,9)(12,13)(15,16,20,23)(17,18,25,21)(19,27,29,22)(24,30,28,26);(8,9)(10,14)(15,17,20,25)(16,21,23,18)(19,28,29,24)(22,26,27,30);(3,4,5,6)(7,8,11,9)(10,12,14,13)(15,17)(16,18)(19,24)(20,25)(21,23)(22,26)(27,30)(28,29);(7,9,11,8)(10,13,14,12)(15,18)(16,17)(19,26)(20,21)(22,24)(23,25)(27,28)(29,30);(7,10)(8,12)(9,13)(11,14)(15,19,20,29)(16,22,23,27)(17,24,25,28)(18,26,21,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609879);