Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8327 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,29)(22,33,35,26)(23,36)(24,28)(25,31)(27,38)(30,44,46,43)(32,42)(34,41,50,48)(37,40)(39,45,47,49);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,24,32)(20,25,28,40)(22,30,35,46)(23,37,38,31)(26,39,33,47)(27,42,36,29)(34,49,50,45)(41,44,48,43);(5,11)(8,14)(12,17)(15,18)(19,22,24,35)(20,26,28,33)(21,30,32,46)(23,34,38,50)(25,39,40,47)(27,41,36,48)(29,43,42,44)(31,45,37,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92171);