Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30839 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 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);(3,4,7,10)(5,6,9,8)(11,12,15,19)(13,21,22,18)(14,23,24,26)(16,25,17,20);(3,5,7,9)(4,8,10,6)(11,13,15,22)(12,18,19,21)(14,20,24,25)(16,23,17,26);(3,6,7,8)(4,9,10,5)(11,14)(12,13)(15,24)(16,21)(17,18)(19,22)(20,26)(23,25);(4,10)(5,9)(11,13)(12,14)(15,22)(16,26)(17,23)(18,25)(19,24)(20,21)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409242);