Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#42964 |
| 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 | |
7086 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);(4,6)(15,16)(17,22)(18,26)(19,24)(20,21)(23,29)(25,28)(27,30);(7,8,11,9)(10,13,14,12)(16,21)(19,22)(23,26)(28,30);(3,4)(5,6)(8,9)(12,13)(15,17,20,24)(16,22,21,19)(18,27,29,25)(23,28,26,30);(7,9,11,8)(10,13,14,12)(15,18,20,29)(16,23,21,26)(17,25,24,27)(19,28,22,30);(7,10,11,14)(8,12,9,13)(15,19,20,22)(16,24,21,17)(18,28,29,30)(23,27,26,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548887);