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×#31163 |
| 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 | |
2226 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)(9,11)(10,12)(13,14)(15,16,20,22,18,21,19,24)(17,25,28,29,26,30,27,23);(3,4)(5,6)(7,9,11,8)(10,13,14,12)(16,21)(17,26)(22,24)(27,28);(4,6)(7,10,11,14)(8,12,9,13)(16,22)(19,20)(21,24)(23,30)(25,29)(27,28);(8,9)(12,13)(15,17,18,26)(16,23,21,29)(19,28,20,27)(22,30,24,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410792);