Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(9,13)(11,14)(15,16,20,17)(18,23,26,21)(19,22,27,24)(25,29,30,28);(4,6)(7,8,10,12)(9,14,13,11)(16,17)(21,23)(22,24)(28,29);(3,4)(5,6)(7,9,10,13)(8,11,12,14)(15,17,20,16)(18,23,26,21)(19,24,27,22)(25,29,30,28);(15,18,20,26)(16,21,17,23)(19,25,27,30)(22,28,24,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,19,20,27)(16,22,17,24)(18,25,26,30)(21,28,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542325);