Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#52301 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C2×#2679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,16)(15,18)(19,20,22,24)(21,25,26,23);(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,18)(16,17)(19,21)(20,23)(22,26)(24,25);(12,16)(15,17)(21,26)(23,25);(4,7)(8,10)(13,18)(15,17)(20,24)(21,26);(3,5,6,9)(4,8,7,10)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540824);