Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
800 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)(5,11,13,8)(6,9)(7,10)(12,17,14,16)(15,18)(19,20)(21,31)(22,28)(23,38)(24,40)(25,30)(26,33)(27,36)(29,39)(32,48)(34,42)(35,45)(37,47)(41,46)(43,50)(44,49);(3,5)(4,8)(6,14)(7,13)(9,17)(10,11)(12,15)(16,18)(19,21,25,35)(20,26,30,44)(22,33,37,49)(23,32,36,50)(24,41,39,34)(27,45,38,31)(28,46,47,42)(29,48,40,43);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22,24,36,25,37,39,23)(20,27,29,28,30,38,40,47)(21,32,34,49,35,50,41,33)(26,42,43,31,44,46,48,45)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389401);