Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35679 |
| 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 | |
9154 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)(7,8,11,9)(10,13,14,12)(15,16,20,17)(18,21,26,23)(19,22,27,24)(25,28,30,29);(3,4)(5,6)(7,9,11,8)(10,13,14,12)(16,17)(19,27)(21,23)(25,30);(7,10)(8,12)(9,13)(11,14)(15,17,20,16)(18,21,26,23)(19,24,27,22)(25,28,30,29);(15,18,20,26)(16,21,17,23)(19,25,27,30)(22,28,24,29);(15,19)(16,22)(17,24)(18,25)(20,27)(21,28)(23,29)(26,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536310);