Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7210 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)(7,8,11,13)(9,10,12,14)(15,16,20,21)(17,22,24,19)(18,26,28,23)(25,27,30,29);(7,9)(8,10)(11,12)(13,14)(16,21)(17,24)(23,26)(25,30);(3,4,5,6)(7,10)(8,12)(9,13)(11,14)(15,17)(16,22)(18,25)(19,21)(20,24)(23,29)(26,27)(28,30);(15,18,20,28)(16,23,21,26)(17,25,24,30)(19,27,22,29);(7,10)(8,12)(9,13)(11,14)(15,19,20,22)(16,17,21,24)(18,27,28,29)(23,25,26,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542523);