Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26491 |
| 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 | |
2206 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)(17,23)(18,28)(19,22)(20,30)(21,26)(24,27)(25,29);(3,4)(5,6)(8,11)(12,13)(15,17,20,19,21,27,29,18)(16,22,25,23,26,28,30,24);(4,6)(15,18,20,17,21,19,29,27)(16,23,25,28,26,24,30,22);(7,9,10,14)(8,12,11,13)(15,19,20,27,21,18,29,17)(16,24,25,22,26,23,30,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351857);