Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C22×#2791 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(6,9)(7,11)(13,14)(15,19)(16,17)(18,20)(21,22,25,28)(23,27,24,26);(6,9)(10,12)(13,15,17,20)(14,18,16,19);(5,6,8,9)(7,10,11,12)(13,16,17,14)(15,18,20,19)(21,23,25,24)(22,26,28,27);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,18)(17,20)(21,24,25,23)(22,27,28,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540756);