Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39512 |
| 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 | |
5358 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);(3,4,7,6)(5,8,10,9)(11,12)(13,17)(14,18)(15,16)(19,20,23,21)(22,25,26,24);(3,5)(4,8)(6,9)(7,10)(12,16)(13,18)(19,21,23,20)(22,25,26,24);(3,4,7,6)(5,9,10,8)(11,13)(12,14)(15,18)(16,17)(19,22,23,26)(20,24,21,25);(11,14)(12,13)(15,17)(16,18);(3,6,7,4)(5,8,10,9)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544041);