Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45300 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7124 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);(3,4,6,5)(8,11)(12,14)(18,26)(21,27)(24,30)(28,29);(4,5)(15,16,19,22)(17,20,25,23)(18,27,26,21)(24,29,30,28);(7,8,10,11)(9,12,13,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,27)(28,30);(3,5,6,4)(7,9,10,13)(8,12,11,14)(15,17)(16,20)(18,24)(19,25)(21,28)(22,23)(26,30)(27,29);(15,18,19,26)(16,21,22,27)(17,24,25,30)(20,28,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577181);