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×#42573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4822 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);(13,17)(16,18)(19,20)(21,26)(22,30)(23,28)(24,25)(27,33)(29,32)(31,34);(3,4)(5,9)(6,7)(8,10)(11,12,15,14)(13,16,17,18)(20,25)(23,26)(27,30)(32,34);(4,7)(8,9)(11,13)(12,16)(14,18)(15,17)(19,21,24,28)(20,26,25,23)(22,31,33,29)(27,32,30,34);(3,4)(5,8)(6,7)(9,10)(11,14,15,12)(13,18,17,16)(19,22,24,33)(20,27,25,30)(21,29,28,31)(23,32,26,34);(3,5,6,10)(4,8,7,9)(19,23,24,26)(20,28,25,21)(22,32,33,34)(27,31,30,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548271);