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×#42814 |
| 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 | |
7150 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)(8,10)(13,14);(7,8,12,10)(9,13,11,14)(15,16,19,22)(17,20,25,23)(18,26,27,21)(24,29,30,28);(3,4,5,6)(7,9,12,11)(8,13,10,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,27)(28,30);(7,10,12,8)(9,13,11,14)(15,17)(16,20)(18,24)(19,25)(21,28)(22,23)(26,29)(27,30);(7,11,12,9)(8,13,10,14)(15,18,19,27)(16,21,22,26)(17,24,25,30)(20,28,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548656);