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×#40039 |
| 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 | |
5398 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);(12,15)(16,18)(19,20,24,21)(22,27,30,25)(23,26,31,28)(29,33,34,32);(4,8)(6,9)(12,15)(13,17)(20,21)(25,27)(26,28)(32,33);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,21,24,20)(22,27,30,25)(23,28,31,26)(29,33,34,32);(3,5,7,10)(4,9,8,6)(19,22,24,30)(20,25,21,27)(23,29,31,34)(26,32,28,33);(3,6,7,9)(4,10,8,5)(11,13,14,17)(12,16,15,18)(19,23,24,31)(20,26,21,28)(22,29,30,34)(25,32,27,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544756);