Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3302 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7982 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);(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,23)(17,25)(18,21)(20,22)(24,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25);(7,12)(9,11)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483483);