Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,9,13)(5,14,6,17)(7,12,8,11)(10,16,18,15)(19,20,25,29)(21,30,33,49)(22,27,24,39)(23,35,38,47)(26,41,44,32)(28,34,46,37)(31,48,40,45)(36,42,50,43);(3,5,9,6)(4,10,13,18)(7,16,8,15)(11,17,12,14)(19,21,25,33)(20,26,29,44)(22,31,24,40)(23,36,38,50)(27,42,39,43)(28,45,46,48)(30,47,49,35)(32,34,41,37);(3,6,9,5)(4,10,13,18)(7,15,8,16)(11,17,12,14)(19,22,25,24)(20,27,29,39)(21,31,33,40)(23,34,38,37)(26,42,44,43)(28,35,46,47)(30,48,49,45)(32,36,41,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396312);