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×#41835 |
| 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 | |
5050 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);(3,4,7,5)(6,8,10,9)(13,17)(16,18)(19,20,24,23)(21,25,29,28)(22,30,32,26)(27,34,31,33);(3,4,7,5)(6,9,10,8)(11,12,15,14)(13,16,17,18)(20,23)(25,28)(26,30)(33,34);(4,5)(8,9)(11,13)(12,16)(14,18)(15,17)(19,21,24,29)(20,25,23,28)(22,31,32,27)(26,34,30,33);(3,5,7,4)(6,9,10,8)(11,12,15,14)(13,16,17,18)(19,22,24,32)(20,26,23,30)(21,27,29,31)(25,33,28,34);(3,6,7,10)(4,8,5,9)(11,14,15,12)(13,18,17,16)(19,23,24,20)(21,28,29,25)(22,30,32,26)(27,34,31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546909);