Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#3218 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(5,7,8,12)(6,9,10,11)(14,18)(19,27)(20,25)(21,28);(7,12)(9,11)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(6,10)(9,11)(14,18)(19,27)(20,25)(21,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482685);