Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,16)(14,15)(17,18)(19,20,23,27,22,26,31,25)(21,28,30,33,29,24,32,34);(3,5,6,10)(4,7,8,9)(12,15)(13,17)(19,21)(20,24)(22,29)(23,30)(25,33)(26,28)(27,34)(31,32);(11,13,14,17)(12,16,15,18)(20,25)(23,31)(24,33)(26,27)(28,34)(30,32);(4,8)(7,9)(12,15)(16,18)(20,26)(21,29)(25,27)(30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368509);