Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#548 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4390 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);(4,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,23)(18,29)(19,24)(20,26)(21,25)(22,27)(28,30);(7,9,11,10)(8,12,14,13)(15,17,19,27)(16,22,24,23)(18,21,28,20)(25,29,26,30);(3,4,5,6)(7,9,11,10)(8,13,14,12)(15,18,21,27,19,28,20,17)(16,23,26,30,24,22,25,29);(7,10,11,9)(8,13,14,12)(16,24)(22,23)(25,26)(29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353853);