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×#26894 |
| 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 | |
2330 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)(15,16,19,24,21,25,20,23)(17,26,27,30,18,29,28,22);(3,4)(5,6)(8,11)(12,13)(15,17,21,18)(16,22,25,30)(19,28,20,27)(23,26,24,29);(4,6)(15,18,21,17)(16,22,25,30)(19,27,20,28)(23,26,24,29);(7,9,10,14)(8,12,11,13)(16,23)(19,20)(22,26)(24,25)(27,28)(29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346825);