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×#59519 |
| 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 | |
5150 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);(3,4,8,6)(5,7,10,9)(12,16)(13,18);(3,5,8,10)(4,9,6,7)(12,16)(13,18)(21,25)(23,26);(11,12,15,16)(13,17,18,14)(19,20,22,24)(21,23,25,26);(3,6,8,4)(5,7,10,9)(11,13)(12,17)(14,16)(15,18)(19,21)(20,23)(22,25)(24,26);(3,7)(4,5)(6,10)(8,9)(11,14,15,17)(12,13,16,18)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573910);