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×#46136 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8,12,13)(9,10,11,14)(15,16,20,23)(17,18,24,21)(19,22,27,29)(25,26,30,28);(8,13)(9,11)(17,24)(18,21)(19,27)(22,29);(3,4)(5,6)(7,9,12,11)(8,14,13,10)(15,17)(16,21)(18,23)(19,25)(20,24)(22,28)(26,29)(27,30);(7,10)(8,9)(11,13)(12,14)(15,18,20,21)(16,17,23,24)(19,26,27,28)(22,25,29,30);(7,11,12,9)(8,10,13,14)(15,19)(16,22)(17,25)(18,26)(20,27)(21,28)(23,29)(24,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610004);