Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50692 |
| 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 | |
6978 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)(15,16,20,25)(17,22,27,21)(18,23,28,29)(19,26,30,24);(7,8)(9,11)(10,13)(12,14)(15,17,20,27)(16,21,25,22)(18,26,28,24)(19,29,30,23);(3,4,5,6)(8,9)(12,13)(15,17)(16,22)(18,24)(19,23)(20,27)(21,25)(26,28)(29,30);(7,9,11,8)(10,13,14,12)(15,18)(16,23)(17,26)(19,21)(20,28)(22,30)(24,27)(25,29);(7,10)(8,12)(9,13)(11,14)(15,19,20,30)(16,24,25,26)(17,23,27,29)(18,21,28,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613698);