Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46202 |
| 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 | |
5012 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);(3,4,8,10)(5,9,6,7)(12,15)(16,18)(19,20,23,26)(21,27,29,24)(22,30,32,25)(28,33,31,34);(5,6)(7,9)(19,21,23,29)(20,24,26,27)(22,31,32,28)(25,34,30,33);(3,5,8,6)(4,7,10,9)(11,12,14,15)(13,16,17,18)(19,21)(20,24)(22,28)(23,29)(25,33)(26,27)(30,34)(31,32);(3,6,8,5)(4,9,10,7)(19,20)(21,24)(22,25)(23,26)(27,29)(28,33)(30,32)(31,34);(3,7,8,9)(4,5,10,6)(11,13,14,17)(12,16,15,18)(19,22,23,32)(20,25,26,30)(21,28,29,31)(24,33,27,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609968);