Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26895 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,10,11)(9,13,14,12)(15,16,19,24,20,25,30,23)(17,26,27,22,18,28,29,21);(3,4)(5,6)(8,11)(12,13)(15,17,20,18)(16,21,25,22)(19,29,30,27)(23,26,24,28);(4,6)(15,18,20,17)(16,22,25,21)(19,27,30,29)(23,28,24,26);(7,9,10,14)(8,12,11,13)(16,23)(19,30)(21,26)(22,28)(24,25)(27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346834);