Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41626 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(8,10)(12,14)(13,17)(19,20,24,28)(21,29,30,25)(22,26,32,34)(23,27,33,31);(11,12,15,14)(13,18,17,16)(21,30)(23,33)(25,29)(27,31);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,16,14,18)(19,21)(20,25)(22,27)(23,26)(24,30)(28,29)(31,32)(33,34);(11,14,15,12)(13,18,17,16)(19,22)(20,26)(21,31)(23,29)(24,32)(25,33)(27,30)(28,34);(3,5,6,9)(4,8,7,10)(19,23,24,33)(20,27,28,31)(21,26,30,34)(22,29,32,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547409);