Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#27593 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2546 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)(11,12,14,16)(13,15,17,18)(19,20)(21,25)(22,27)(23,32)(24,28)(26,29)(30,34)(31,33);(3,5,6,10)(4,7,8,9)(11,13)(12,15)(14,17)(16,18)(19,21)(20,25)(22,31)(23,29)(24,30)(26,32)(27,33)(28,34);(13,17)(15,18)(19,21)(20,26)(22,29)(23,31)(24,30)(25,32)(27,34)(28,33);(4,8)(7,9)(19,22,24,23)(20,27,28,32)(21,29,30,31)(25,26,34,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365986);