Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29848 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,10,11)(9,12,13,14)(15,16,19,23,20,18,21,24)(17,25,28,29,27,22,26,30);(3,4)(5,6)(8,11)(12,14)(15,17)(16,22)(18,25)(19,28)(20,27)(21,26)(23,30)(24,29);(4,6)(9,13)(12,14)(15,18,20,16)(17,22,27,25)(19,23,21,24)(26,30,28,29);(7,9,10,13)(8,12,11,14)(15,19,20,21)(16,23,18,24)(17,26,27,28)(22,29,25,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342731);