Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36027 |
| 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 | |
5160 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);(4,5)(8,9)(12,16)(17,18)(19,20,24,22)(21,27,28,25)(23,26,31,30)(29,33,34,32);(3,4,7,5)(6,8,10,9)(21,28)(25,27)(29,34)(32,33);(3,5,7,4)(6,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,21)(20,25)(22,27)(23,29)(24,28)(26,32)(30,33)(31,34);(11,13,15,14)(12,17,16,18)(19,22,24,20)(21,27,28,25)(23,26,31,30)(29,32,34,33);(3,6,7,10)(4,8,5,9)(11,14,15,13)(12,18,16,17)(19,23,24,31)(20,26,22,30)(21,29,28,34)(25,32,27,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536733);