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×#35311 |
| 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 | |
9174 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)(9,13)(11,14)(15,16,19,22)(17,20,25,23)(18,26,27,21)(24,29,30,28);(3,4)(5,6)(7,8,10,12)(9,11,13,14)(15,16,19,22)(17,23,25,20)(18,21,27,26)(24,29,30,28);(7,9)(8,11)(10,13)(12,14)(15,16)(17,20)(18,21)(19,22)(23,25)(24,28)(26,27)(29,30);(15,17)(16,20)(18,24)(19,25)(21,28)(22,23)(26,29)(27,30);(15,18,19,27)(16,21,22,26)(17,24,25,30)(20,28,23,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535789);