Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(6,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,28)(22,25)(23,26)(24,27);(13,15,16,22)(14,19,18,25)(17,24,23,21)(20,27,26,28);(5,7,8,11)(6,10,9,12)(14,18)(19,25)(20,26)(27,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482281);