Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31243 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 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);(7,8,11,13)(9,10,14,12)(15,16,21,24,20,26,19,25)(17,22,29,30,18,28,27,23);(3,4)(5,6)(7,9,11,14)(8,12,13,10)(15,17,20,18)(16,22,26,28)(19,27,21,29)(23,24,30,25);(4,6)(15,18,20,17)(16,23,26,30)(19,27,21,29)(22,24,28,25);(7,10,11,12)(8,9,13,14)(16,24)(19,21)(22,30)(23,28)(25,26)(27,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411068);