Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7750 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)(8,12)(10,14)(15,16)(17,23)(18,25)(19,27)(20,21)(22,28)(24,26)(29,30);(8,12)(9,13)(16,21)(18,23)(22,27)(26,29);(3,4,5,6)(7,8,11,12)(9,10,13,14)(15,17)(16,18)(19,24)(20,25)(21,23)(22,26)(27,29)(28,30);(7,9,11,13)(8,10,12,14)(15,18)(16,17)(19,26)(20,23)(21,25)(22,24)(27,30)(28,29);(7,10)(8,13)(9,12)(11,14)(15,19,20,28)(16,22,21,27)(17,24,25,30)(18,26,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541691);