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×#39465 |
| 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 | |
8208 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)(7,8,11,14)(9,10,13,12)(16,20)(17,23)(22,25)(24,29);(9,13)(10,12)(15,16,19,20)(17,21,23,27)(18,25,26,22)(24,30,29,28);(3,4)(5,6)(7,9,11,13)(8,12,14,10);(15,17)(16,21)(18,24)(19,23)(20,27)(22,28)(25,30)(26,29);(7,10,11,12)(8,13,14,9)(15,18,19,26)(16,22,20,25)(17,24,23,29)(21,28,27,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544047);