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×#45856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4984 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);(3,4,8,9)(5,6,7,10)(11,12,14,16)(13,15,17,18)(21,27)(25,30)(28,33)(32,34);(4,9)(5,7)(20,24)(21,27)(22,29)(32,34);(3,5,8,7)(4,10,9,6)(11,12)(13,15)(14,16)(17,18)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34);(3,6)(4,5)(7,9)(8,10)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,28)(23,27)(24,30)(26,32)(29,33)(31,34);(3,7,8,5)(4,6,9,10)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609657);