Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(7,8)(9,12)(10,11)(13,14)(15,16,19,22)(17,20,24,28)(18,21,25,29)(23,27,26,30);(3,5,6,4)(8,11)(12,14)(17,23)(18,25)(20,27)(21,29)(24,26)(28,30);(4,5)(7,9,10,13)(8,12,11,14)(15,17)(16,20)(18,26)(19,24)(21,30)(22,28)(23,25)(27,29);(8,11)(12,14)(17,24)(20,28)(23,26)(27,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348800);