Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#23718 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,18,15,20,14,19)(13,17,22,25,23,26,21,24);(4,7)(8,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,25)(19,24)(20,26);(12,18)(13,21)(14,16)(19,20)(22,23)(24,25);(3,5,6,9)(4,8,7,10)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339549);