Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2026 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);(4,8,10,13)(9,17,18,16)(19,20,23,28,24,27,38,42)(21,29,32,43,31,25,36,40)(22,33,35,44,34,26,37,41)(30,45,47,50,46,39,48,49);(3,4)(5,9)(6,13)(7,10)(8,15)(11,16)(12,18)(14,17)(19,21)(20,25)(22,30)(23,36)(24,31)(26,39)(27,29)(28,43)(32,38)(33,45)(34,46)(35,48)(37,47)(40,42)(41,50)(44,49);(3,5,7,12)(4,9,10,18)(6,14,15,11)(8,16,13,17)(19,22)(20,26)(21,30)(23,37)(24,34)(25,39)(27,33)(28,44)(29,45)(31,46)(32,48)(35,38)(36,47)(40,50)(41,42)(43,49)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94421);