Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,8)(6,10)(12,15)(13,17)(19,20,23,25)(21,22,24,26);(3,4,7,8)(5,6,9,10)(12,15)(13,17)(19,21)(20,22)(23,24)(25,26);(3,5,7,9)(4,6,8,10)(11,12)(13,16)(14,15)(17,18);(11,13,14,17)(12,16,15,18)(19,22)(20,24)(21,25)(23,26);(3,6)(4,9)(5,8)(7,10)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573653);