Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8650 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)(7,8,12,11)(9,14,13,10)(15,16)(17,20)(18,21)(19,24)(22,27)(23,25)(26,29)(28,30);(7,9)(8,10)(11,14)(12,13)(16,17)(18,25)(22,24)(26,30);(3,4)(5,6)(7,10,12,14)(8,13,11,9)(15,17,20,16)(18,23,25,21)(19,24,27,22)(26,29,30,28);(7,11,12,8)(9,14,13,10)(15,18,20,25)(16,21,17,23)(19,26,27,30)(22,28,24,29);(15,19,20,27)(16,22,17,24)(18,26,25,30)(21,28,23,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535787);