Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3257 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7078 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);(6,9)(7,11)(13,14)(15,18)(16,19)(17,21)(20,24)(22,26)(23,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(6,9)(10,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(5,7,8,11)(6,10,9,12)(15,22)(18,26)(23,28)(25,27)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482802);