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×#46283 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,8,10)(5,9,6,7)(11,12,14,16)(13,17,18,15)(19,20)(21,27)(22,30)(23,26)(24,29)(25,31)(28,32)(33,34);(5,6)(7,9)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(3,5,8,6)(4,7,10,9)(11,13,14,18)(12,15,16,17)(19,20)(21,24)(22,25)(23,26)(27,29)(28,32)(30,31)(33,34);(3,6,8,5)(4,9,10,7)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(3,7,8,9)(4,5,10,6)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610181);