Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17502 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3538 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);(3,4,7,9,8,5,6,10)(11,12)(13,21)(14,20)(15,18)(16,19)(17,22)(23,26)(24,25);(3,5)(4,8)(6,9)(7,10)(11,13)(12,17)(14,22)(15,24)(16,23)(18,21)(19,25)(20,26);(3,6,8,7)(4,9,5,10)(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,21,25,26);(12,19)(17,25)(18,20)(21,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250205);