Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55670 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4990 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)(6,9,10,8)(12,14)(13,15)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,33)(32,34);(4,5)(8,9)(11,12)(13,17)(14,16)(15,18)(19,20,23,26)(21,27,29,24)(22,30,31,25)(28,32,34,33);(11,13,16,15)(12,17,14,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(3,5,7,4)(6,9,10,8)(11,14,16,12)(13,17,15,18)(19,21,23,29)(20,24,26,27)(22,28,31,34)(25,32,30,33);(3,6,7,10)(4,8,5,9)(11,15,16,13)(12,17,14,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008925);