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×#39379 |
| 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 | |
6974 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)(15,16,20,19)(17,21,25,24)(18,26,27,22)(23,29,30,28);(4,5)(7,8,10,12)(9,11,13,14)(16,19)(21,24)(22,26)(28,29);(3,4,6,5)(7,9,10,13)(8,11,12,14)(15,17,20,25)(16,21,19,24)(18,23,27,30)(22,28,26,29);(15,18,20,27)(16,22,19,26)(17,23,25,30)(21,28,24,29);(3,5,6,4)(15,19,20,16)(17,24,25,21)(18,26,27,22)(23,29,30,28)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543931);